# Generalized Differentiability of Continuous Functions

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. General Definitions and Conventions

**Definition**

**1.**

**Definition**

**2**

## 3. The Discontinuity Sets of Functions

**Definition**

**3**

**Definition**

**4**

**Lemma**

**1**

**Corollary**

**1.**

**Theorem**

**1**

**Proof.**

**Increasing case**: Suppose that F is increasing in ${I}_{n-1}$ then

**Decreasing case**: Suppose that F is decreasing in ${I}_{n-1}$ then

## 4. Indicial $\omega $ Derivatives

**Definition**

**5**

- (1)
- non-decreasing continuous function, such that
- (2)
- ${g}_{x}\left(0\right)=0$ and
- (3)
- $|{\mathrm{\Delta}}_{\u03f5}^{\pm}\left[f\right]\left(x\right)|\le K\phantom{\rule{4pt}{0ex}}{g}_{x}\left(\u03f5\right)$ holds in the interval $I=[x,x\pm \u03f5]$ for some constant $K>0$.

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Remark**

**1.**

**Theorem**

**2**

**Proof.**

**Definition**

**9.**

**Theorem**

**3**

**Proof.**

**Definition**

**10.**

**Theorem**

**4**

**Proof.**

**Theorem**

**5**

**Proof.**

**Continuity implication**Consider the inequality

**Forward statement**: Suppose that ${\overline{\mathcal{D}}}_{\omega}^{\pm}f\left(x\right)={L}_{1}$ and ${\underset{\overline{}}{\mathcal{D}}}_{\omega}^{\pm}f\left(x\right)={L}_{2}$ Then by LUB

**Converse statement**: Suppose that

**Corollary**

**2**

**Proof.**

**Theorem**

**6**

**Proof.**

## 5. Fat Cantor Sets and Related Quasi-Singular Functions

**Definition**

**11**

- f is non-constant on I.
- the complement ${\mathcal{N}}^{c}\cap I$ is totally disconnected.

**Definition**

**12**

**Theorem**

**7.**

**Corollary**

**3.**

## 6. Continuity Sets of Derivatives

**Theorem**

**8**

**Proof.**

**Definition**

**13.**

**Theorem**

**9**

**Proof.**

**Corollary**

**4.**

**Theorem**

**10**

- (1)
- ${f}_{+}^{\prime}\left(x\right)={f}_{-}^{\prime}\left(x\right)={f}^{\prime}\left(x\right)$
- (2)
- ${\mathrm{\Delta}}_{f,I}:=\{x:{f}^{\prime}\notin \mathcal{C},x\in I\}$ is totally disconnected with empty interior.
- (3)
- The total discontinuity set can be written as ${\mathrm{\Delta}}_{f,I}={\mathrm{\Delta}}_{1,f}\cup {\mathrm{\Delta}}_{2,f}$, where ${\mathrm{\Delta}}_{1,f}$ is ${F}_{\sigma}$ and ${\mathrm{\Delta}}_{2,f}$ is a null set.
- (4)
- The continuity set ${\mathcal{C}}_{f}$ is ${G}_{\delta}$. ${F}_{\sigma}$ and ${G}_{\delta}$ are given by Definition A4

**Proof.**

## 7. Characterization of Fractional Velocities

**Definition**

**14.**

**Definition**

**15**

**Remark**

**2.**

**Definition**

**16.**

**Proposition**

**1.**

**Proof.**

## 8. Discussion

## Funding

## Conflicts of Interest

## Appendix A. Additional Notations

**Definition**

**A1**

**Definition**

**A2**

**null set**$Z\subset {\mathbb{R}}^{\phantom{\rule{0.166667em}{0ex}}}$ (or a set of measure 0) is called a set, such that for every $0<\u03f5<1$ there is a countable collection of sub-intervals ${\left(\right)}_{{I}_{k}}^{}$, such that

**Definition**

**A3**

**Definition**

**A4**

- The set $E\subseteq X$ is ${G}_{\delta}$ if it is countable intersection of open sets, and it is ${F}_{\sigma}$ if it is countable union of closed sets.
- The set $E\subseteq X$ is meagre if it can be expressed as the union of countably many nowhere dense subsets of X.
- Dually, a co-meagre set is one whose complement is meagre, or equivalently, the intersection of countably many sets with dense interiors.

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Prodanov, D.
Generalized Differentiability of Continuous Functions. *Fractal Fract.* **2020**, *4*, 56.
https://doi.org/10.3390/fractalfract4040056

**AMA Style**

Prodanov D.
Generalized Differentiability of Continuous Functions. *Fractal and Fractional*. 2020; 4(4):56.
https://doi.org/10.3390/fractalfract4040056

**Chicago/Turabian Style**

Prodanov, Dimiter.
2020. "Generalized Differentiability of Continuous Functions" *Fractal and Fractional* 4, no. 4: 56.
https://doi.org/10.3390/fractalfract4040056