# 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 ${\left\{{I}_{k}\right\}}_{k=1}^{\infty}$, 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.

## References

- Ampére, A.M. Recherches sur quelques points de la théorie des fonctions dérivées qui condiusent à une nouvelle démonstration de la série de Taylor, et à l’expression finie des termes qu’on néglige lorsqu’on arrête cette série à unterme quelconque. J. Ecole Polytech.
**1806**, 6, 148–181. [Google Scholar] - Kucharski, A. Math’s beautiful monsters. Nautilus
**2014**, 3, 11. [Google Scholar] - Mandelbrot, B. Fractal Geometry of Nature; W.H. Freeman: San Francisco, CA, USA, 1982. [Google Scholar]
- Prodanov, D. Characterization of strongly non-linear and singular functions by scale space analysis. Chaos Solitons Fractals
**2016**, 93, 14–19. [Google Scholar] [CrossRef][Green Version] - Prodanov, D. Fractional velocity as a tool for the study of non-linear problems. Fractal Fract.
**2018**, 2, 4. [Google Scholar] [CrossRef][Green Version] - Milanov, S.; Petrova-Deneva, A.; Angelov, A.; Shopolov, N. Higher Mathematics, Part II; Technika: Sofia, Bulgaria, 1977. [Google Scholar]
- Parvate, A.; Gangal, A. Calculus on fractal subsets of real line—I: Formulation. Fractals
**2009**, 17, 53–81. [Google Scholar] [CrossRef] - Parvate, A.; Satin, S.; Gangal, A. Calculus on fractal curves in r
^{n}. Fractals**2011**, 19, 15–27. [Google Scholar] [CrossRef][Green Version] - Yang, X.-J.; Baleanu, D.; Srivastava, H. Local Fractional Integral Transforms and Their Applications; Academic Press: London, UK, 2015. [Google Scholar]
- Cherbit, G. Fractals, dimension non entiére et applications. In Dimension Locale, Quantité de Mouvement et Trajectoire; Masson: Paris, France, 1987; pp. 340–352. [Google Scholar]
- Prodanov, D. Conditions for continuity of fractional velocity and existence of fractional taylor expansions. Chaos Solitons Fractals
**2017**, 102, 236–244. [Google Scholar] [CrossRef] - Prodanov, D. Characterization of the local growth of two Cantor–type functions. Fractal Fract.
**2019**, 3, 45. [Google Scholar] [CrossRef][Green Version] - Brown, J.; Darji, U.; Larsen, E. Nowhere monotone functions and functions of non-monotonic type. Proc. Am. Math. Soc.
**1999**, 127, 173–182. [Google Scholar] [CrossRef] - Prodanov, D. Fractional variation of Hölderian functions. Fract. Calc. Appl. Anal.
**2015**, 18, 580–602. [Google Scholar] [CrossRef][Green Version] - Bartle, R. Modern Theory of Integration, Volume 32 of Graduate Studies in Mathematics; American Mathematical Society: Providence, RI, USA, 2001. [Google Scholar]

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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