The Specular Derivative Kiyuob Jung Jehan Oh Abstract

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The Specular Derivative

Kiyuob Jung∗

, Jehan Oh†

Abstract

In this paper, we introduce a new generalized derivative, which we term the specular derivative. We establish

the Quasi-Rolles’ Theorem, the Quasi-Mean Value Theorem, and the Fundamental Theorem of Calculus in light

of the specular derivative. We also investigate various analytic and geometric properties of specular derivatives

and apply these properties to several diﬀerential equations.

Key words: generalization of derivatives, Fundamental Theorem of Calculus, Quasi-Mean Value Theorem,

tangent hyperplanes, diﬀerential equations

AMS Subject Classiﬁcations: 26A24, 26A27, 26B12, 34A36

1 Introduction

A derivative is a fundamental tool to measure the change of real-valued functions. The application of derivatives

has been investigated in diverse ﬁelds beyond mathematics. Simultaneously, the generalization of derivatives has

been studied in the ﬁelds of mathematical analysis, complex analysis, algebra, and geometry. The reason why

we investigate to generalize derivatives is that the condition, such as continuity or smoothness or measurability,

required to have diﬀerentiability is demanding. In this sense, we devote this paper to device a way to generalize a

derivative in accordance with our intuition and knowledge.

Let fbe a single-variable function deﬁned on an interval Iin Rand let xbe a point in I. In order to avoid

confusion we call f0aclassical derivative in this paper. When we say to generalize a derivative, the precise meaning

is to ﬁnd an operator which a classical derivative implies. Also, generalization of derivatives includes the relation-

ship with Riemann or Lebesgue integration. There is a lot of ways to achieve this task: symmetric derivatives,

subderivatives, weak derivatives, Dini derivatives, and so on. Extensive and well-organized survey for the foregoing

discussion can be found in [6] and [5]. Subderivatives are motivated the geometric properties of the tangent line with

classical derivatives. The concept of subderivatives can be deﬁned in abstract function spaces. Weak derivatives are

motivated by the integration by parts formula and is related to the functional analysis. We are interested in the

application of generalized derivatives to partial diﬀerential equations and refer to Evans [8] and Bressan [4].

In order to refer to our study, we look over symmetric derivatives, denoted by f∗, made by changing the form

of the diﬀerent quotient. Since f∗(x) dose not depend on the behavior of fat the point x, the symmetric derivative

f∗(x) can exist even if f(x) does not exist. Note that the existence of f∗(x) does not imply the existence of f0(x).

However, if f0(x) exists almost everywhere, then f∗(x) exists almost everywhere. If fand f∗are continuous on

an open interval I, then there exists x∈Isuch that f0(x) = f∗(x). Symmetric derivatives do not satisfy the

classical Rolle’s Theorem and Mean Value Theorem. As replacements so-called the Quasi-Rolle’s Theorem and the

Quasi-Mean Value Theorem for continuous functions was proved by Aull [1]. Larson [9] proved that the continuity

in Quasi-mean value theorem can be replaced by measurability. As for Quasi-mean value theorem for symmetric

derivatives, [11] and [10] can be not only accessible but also extensive. Furthermore, according to Aull [1], the

Quasi-Mean Value Theorem implies that symmetric derivatives satisfy the property akin to a Lipschitz condition.

In this paper, we device a new generalized derivative so-called a specular derivative including the classical

derivative in not only one-dimensional space Rbut also high-dimensional space Rn. Also, we examine various

analytic and geometric properties of specular derivatives and apply these properties in order to address several

diﬀerential equations.

To give an intuition, consider a function fwhich is continuous at x0but not diﬀerentiable at x0in Ias in Figure

1. Imagine that you shot a light ray from left to right toward a certain mirror and then the light ray makes a turn

along at the point x0. The light ray can be represented as two lines T1with the right-hand derivative f0

+(x0) and

T2with the left-hand derivative f0

−(x0) that just touch the function fat the point x0. Finally, the mirror must be

∗Department of Mathematics, Kyungpook National University, Daegu, 41566, Republic of Korea, E-mail: kyjung2357@knu.ac.kr

†Department of Mathematics, Kyungpook National University, Daegu, 41566, Republic of Korea, E-mail: jehan.oh@knu.ac.kr

arXiv:2210.06062v1 [math.CA] 12 Oct 2022

the line T. Moreover, the angle between T1and T is equal with the angle between T2and T. We deﬁne the slope

of the line T as the specular derivative of fat x0, written by f∧(x0). The word “specular” in specular derivatives

stands for the mirror T.

Figure 1: Motivation for specular derivatives

Here are our main results. The specular derivative is well-deﬁned in Rnfor each n∈N. In one-dimensional space

R, we suggest three ways to calculate a specular derivative and prove that specular derivatives obey Quasi-Rolles’

Theorem and Quasi-Mean Value Theorem. Interestingly, the second order specular diﬀerentiability implies the ﬁrst

order classical diﬀerentiability. The most noteworthy is the Fundamental Theorem of Calculus can be generalized

in the specular derivative sense. By deﬁning a tangent hyperplane in light of specular derivatives, we extend the

concepts of specular derivatives in high-dimensional space Rnand provide several examples. Especially, we reveal

that the directional derivative with specular derivatives is related to the gradient with specular derivatives and has

extrema. As for diﬀerential equations, we construct and address the ﬁrst order ordinary diﬀerential equation and

the partial diﬀerential equation, called the transport equation, with specular derivatives.

The rest of the paper is organized as follows. In Section 2, we deﬁne a specular derivative in one-dimensional

space Rand state properties of the specular derivative. Section 3 extends the concepts of the specular derivative

to high-dimensional space Rn. Also, the gradient and directional derivatives for specular derivatives are provided

in Section 3. Section 4 deals with diﬀerential equations with specular derivatives. Starting from the Fundamental

Theorem of Calculus with specular derivatives, Section 4 constructs and solves the ﬁrst order ordinary diﬀerential

equation and the transport equation with specular derivatives. Appendix contains delayed proofs, properties useful

but elementary, and notations comparing classical derivatives and specular derivatives.

2 Specular derivatives for single-variable functions

Here is our blueprint for specular derivatives in one-dimensional space R. In Figure 2, a function fis specularly

diﬀerentiable in a open interval (a, b)⊂Reven if fis not deﬁned at a countable sequence α1,α2,···,αnand is

not diﬀerentiable at some points.

Figure 2: The blueprint for specular derivatives in one-dimension

2.1 Deﬁnitions and properties

Deﬁnition 2.1. Let f:I→Rbe a single-variable function with an open interval I⊂Rand x0be a point in I.

Write

f[x0) := lim

x&x0

f(x) and f(x0] := lim

x%x0

f(x)

if each limit exists. Also, we denote f[x0] := 1

2(f[x0) + f(x0]).

Deﬁnition 2.2. Let f:I→Rbe a function with an open interval I⊂Rand x0be a point in I. We say fis right

specularly diﬀerentiable at x0if x0is a limit point of I∩[x0,∞) and the limit

f∧

+(x0) := lim

x&x0

f(x)−f[x0)

x−x0

exists as a real number. Similarly, we say fis left specularly diﬀerentiable at x0if x0is a limit point of I∩(−∞, x0]

and the limit

f∧

−(x0) := lim

x%x0

f(x)−f(x0]

x−x0

exists as a real number. Also, we call f∧

+(x0) and f∧

−(x0) the (ﬁrst order)right specular derivative of fat x0and the

(ﬁrst order)left specular derivative of fat x0, respectively. In particular, we say fis semi-specularly diﬀerentiable

at x0if fis right and left specularly diﬀerentiable at x0.

In Appendix 5.2, we suggest the notation for semi-specular derivatives and employ the notations in this paper.

Remark 2.3. Clearly, semi-diﬀerentiability implies semi-specular diﬀerentiability, while the converse does not

imply. For example, the sign function is neither right diﬀerentiable nor left diﬀerentiable at 0, whereas one can

prove that DRsgn(0) = 0 = DLsgn(0).

Deﬁnition 2.4. Let Ibe an open interval in Rand x0be a point in I. Suppose a function f:I→Ris semi-

specularly diﬀerentiable at x0. We deﬁne the phototangent of fat x0to be the function pht f:R→Rby

pht f(y) = 









f∧

−(x0)(y−x0) + f(x0] if y < x0,

f[x0] if y=x0,

f∧

+(x0)(y−x0) + f[x0) if y > x0.

Deﬁnition 2.5. Let f:I→Rbe a function, where Iis a open interval in R. Let x0be a point in I. Suppose fis

semi-specularly diﬀerentiable at x0and let pht fbe the phototangent of fat x0. Write x0= (x0, f [x0]) ∈I×R.

(i) The function fis said to be specularly diﬀerentiable at x0if pht fand a circle ∂B (x0, r) have two intersection

points for all r > 0.

(ii) Suppose fis specularly diﬀerentiable at x0and ﬁx r > 0. The (ﬁrst order)specular derivative of fat x0,

denoted by f∧(x0), is deﬁned to be the slope of the line passing through the two distinct intersection points

of pht fand the circle ∂B (x0, r).

In particular, if fis specularly diﬀerentiable on a closed interval [a, b], then we deﬁne specular derivatives at

end points: f∧(a) := f∧

+(a) and f∧(b) := f∧

−(b). We say fis specularly diﬀerentiable on an interval Iin Rif fis

specularly diﬀerentiable at x0for all x0∈I.

Note that specular derivatives are translation-invariant. Also, if fis specularly diﬀerentiable on an interval

I⊂R, then the set of all points at which fhas a removable discontinuity is at most countable since f(x] and f[x)

exist for all x∈I.

Proposition 2.6. Let f:I→Rbe a function for an open interval I⊂Rand x0be a point in I. Suppose

there exists a phototangent, say pht f, of fat x0. Then fis specularly diﬀerentiable at x0if and only if pht fis

continuous at x0.

Proof. Write x0= (x0,pht f(x0)). Let r > 0 be a real number. Write a circle ∂B(x0, r) as the equation:

(x−x0)2+ (y−pht f(x0))2=r2(2.1)

for x, y ∈R. The system of (2.1) and pht f|[x0,∞)has a root aas well as the system of (2.1) and pht f|(−∞,x0]has

a root b:

a:= x0+ x2

0+r2

f∧

+(x0)2+ 1!

and b:= x0− x2

0+r2

f∧

−(x0)2+ 1!

,(2.2)

using the quadratic formula. Notice that b<a.

To prove that pht fis continuous at x0, take δ:= min {a−x0, x0−b}. Then δ > 0. If z∈(x0−δ, x0+δ), then

|f[x0]−pht f(a)|< r if z∈[x0, a),

which implies that pht fis continuous at x0.

Conversely, the system of (2.1) and pht fhas two distinct roots since b < x0< a. Hence, we conclude that fis

specularly diﬀerentiable at x0.

Corollary 2.7. Let fand gbe single valued functions on an open interval I⊂Rcontaining a point x0. Suppose f

and gare specularly diﬀerentiable at x0. Then f+gis specularly diﬀerentiable at x0and pht f+ pht g= pht(f+g).

Example 2.8. The phototangent of the sign function at 0 is itself, which is not continuous at 0. Hence, the sign

function is not specularly diﬀerentiable at 0.

Example 2.9. The function f:R→Rby f(x) = |x|for x∈Ris continuous and specularly diﬀerentiable on R.

In fact, f∧(x) = sgn(x) which is the sign function. Let g:R→Rbe the function deﬁned by g(x) = |x|if x6= 0

and g(x) = 1 if x= 0. Note that gis not continuous at 0 but is specularly diﬀerentiable at 0 with g∧(0) = 0.

Consequently, we have f∧(x) = g∧(x) = sgn(x) for all x∈R.

Deﬁnition 2.10. Let f:I→Rbe a function with an interval I⊂Rand a point x0in I. Suppose fis specularly

diﬀerentiable at x0. We deﬁne the specular tangent line to the graph of fat the point (x0, f[x0]), denoted by stg f,

to be the line passing through the point (x0, f[x0]) with slope f∧(x0).

Remark 2.11. In Deﬁnition 2.10, the specular tangent line is given by the function stg f:I→Rby

stg f(x) = f∧(x0)(x−x0) + f[x0]

for x∈I. Also, the specular tangent has two properties: f[x0] = stg f(x0) and f∧(x0) = (stg f)∧(x0).

In Figure 3, the function fis neither continuous at x0nor diﬀerentiable at x0. Let pht fbe the phototangent of

fat x0. We can calculate the specular derivative whenever pht fis continuous at x0. Imagine you shot a light ray

toward a mirror. The words “specular” in specular tangent line and “photo” in phototangent stand for the mirror

stg fand the light ray pht f, respectively. Write C = (x0, f[x0]). Observing that

∠CPQ = ∠CQP = ∠SCQ = ∠TCP,

one can ﬁnd that the slope of the line PQ and the slope of the phototangent of fat x0are equal.

Figure 3: Basic concepts concerning specular derivatives

We suggest three avenues calculating a specular derivative. The ﬁrst formula can be used as the criterion for

the existence of specular derivatives.

Theorem 2.12. (Specular Derivatives Criterion) Let f:I→Rbe a function with an open interval I⊂Rand x

be a point in I. If fis specularly diﬀerentiable at x, then

f∧(x) = lim

h→0

(f(x+h)−f[x]) q(f(x−h)−f[x])2+h2−(f(x−h)−f[x]) q(f(x+h)−f[x])2+h2

hq(f(x−h)−f[x])2+h2+hq(f(x+h)−f[x])2+h2

Proof. Set Γ := {y−x:y∈I}and deﬁne the function g: Γ →Rby g(γ) = f(x+γ)−f[x] for γ∈Γ. Since

specular derivatives are translation-invariant, we have f∧(x) = g∧(0). Hence, it suﬃces to prove that

g∧(0) = lim

h→0

g(h)pg(−h)2+h2−g(−h)pg(h)2+h2

hpg(−h)2+h2+hpg(h)2+h2.

Let h > 0 be given. Fix 0 < r < h. Since gis specularly diﬀerentiable at zero, there exists a circle ∂B(O, r)

centered at the origin O with radius r. Moreover, the circle ∂B(0, r) has the two distinct points A and B intersecting

the half-lines −→

OC and −→

OD, respectively, where C = (h, g(h)) and D = (−h, g(−h)). See Figure 4. Observe the similar

right triangles:

4AFO ∼ 4CEO and 4BGO ∼ 4DHO,

where points E = (h, 0), F = (A •e1,0), G = (B •e1,0), and H = (h, 0) with e1= (1,0). One can ﬁnd that

F = rh

pg(h)2+h2,0!and G = −rh

pg(−h)2+h2,0!,

using basic geometry properties for similar right triangles. Consider the continuous function q:R→Rdeﬁned by

q(z) = 









g(h)

hzif z < 0,

0 if z= 0,

g(−h)

hzif z > 0,

which passes through points D, B, O, A, and C. Using the function q, we ﬁnd that

A = rh

pg(h)2+h2,rg(h)

pg(h)2+h2!and B = −rh

pg(−h)2+h2,rg(−h)

pg(−h)2+h2!.

Hence, the slope of the line AB is

g(h)pg(−h)2+h2−g(−h)pg(h)2+h2

hpg(−h)2+h2+hpg(h)2+h2=: σ(h).

Note that θ1=∠AOP and θ2=∠BOQ converges to zero as h→0, where P and Q are the intersection points

of the circle ∂B(O, r) and pht g. The deﬁnition of the specular derivative yields that σ(h) converges to g∧(0) as

h&0. Since the function σis even, we deduce that σ(h) converges to g∧(0) as h→0, completing the proof. 

Figure 4: The slope of the line AB converges the specular derivative of gat zero

In the proof of Theorem 2.12, the observation that the function σis even can be generalized as follows:

Corollary 2.13. Let f:I→Rbe a function with an open interval I⊂R. Let x0be a point in I. Assume fis

specularly diﬀerentiable at x0. If fis symmetric about x=x0in a neighborhood of x0, that is, there exists δ > 0

such that

f(x0−x) = f(x0+x)

for all x∈(x0−δ, x0+δ). Then f∧(x0)=0.

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TheSpecularDerivativeKiyuobJung*,JehanOhAbstractInthispaper,weintroduceanewgeneralizedderivative,whichwetermthespecularderivative.WeestablishtheQuasi-Rolles'Theorem,theQuasi-MeanValueTheorem,andtheFundamentalTheoremofCalculusinlightofthespecularderivative.Wealsoinvestigatevariousanalyticandgeometri...

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