# Characterization of the Local Growth of Two Cantor-Type Functions

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Bounds of Growth of the Cantor’S Function

**Definition**

**1.**

**Upper bond of**$F\left(x\right).$ By the functional Equation (1) and the monotonicity $F\left(x\right)\le {x}^{\alpha}$.**Lower bond**$F\left(x\right).$ For $x=\frac{2}{{3}^{n}}$ we have $F\left(\frac{2}{{3}^{n}}\right)=\frac{1}{{2}^{n}}$. Therefore, since F is increasing,$$F\left(x\right)\ge {\left(\frac{x}{2}\right)}^{\alpha}.$$

## 3. Point-Wise Oscillation of Functions

**Definition**

**2**

**Definition**

**3.**

**Lemma**

**1**

**Proof.**

**Forward case**Suppose that ${\mathrm{osc}}_{}^{+}\left[f\right]\left(x\right)=0$. Then there exists a pair $\mu ::\delta ,\phantom{\rule{4pt}{0ex}}\delta \le \u03f5$, such that ${\mathrm{osc}}_{\delta}^{+}\left[f\right]\left(x\right)\le \mu $. Therefore, f is bounded in ${I}_{+}$. Since $\mu $ is arbitrary, we select ${x}^{\prime}$ such that

**Reverse case.**If f is (right-) continuous at $x\in I=[x,x+\u03f5]$ then there exists a pair $\mu ::\delta $ such that

**Corollary**

**1.**

**Definition**

**4.**

## 4. Fractional Velocity

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7**

**Condition**

**1**

**Condition**

**2**

**Theorem**

**1**

**Proof.**

**Forward statement**

**Converse statement**

## 5. Fractional Velocities of the Cantor Function

**Theorem**

**2.**

**Remark**

**1**

**Remark**

**2**

## 6. The Smith–Volterra–Cantor Set and Its Related Singular Function

**Definition**

**8**

**Theorem**

**3.**

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Recursive construction of Cantor’s function and its bounds. $F\left(x\right)$ is approximated by an iteration function system, $n=8$ iterations. (

**a**) $L\left(x\right)\le F\left(x\right)\le U\left(x\right)$; (

**b**) $1-U(1-x)\le F\left(x\right)\le 1-L(1-x)$.

**Figure 2.**Approximations of the SVC (Smith-Volterra-Cantor) and Cantor’s functions. Blue (discrete1)—SVC function, red (discrete2)—Cantor’s function; both functions are computed for six levels of iteration.

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Prodanov, D.
Characterization of the Local Growth of Two Cantor-Type Functions. *Fractal Fract.* **2019**, *3*, 45.
https://doi.org/10.3390/fractalfract3030045

**AMA Style**

Prodanov D.
Characterization of the Local Growth of Two Cantor-Type Functions. *Fractal and Fractional*. 2019; 3(3):45.
https://doi.org/10.3390/fractalfract3030045

**Chicago/Turabian Style**

Prodanov, Dimiter.
2019. "Characterization of the Local Growth of Two Cantor-Type Functions" *Fractal and Fractional* 3, no. 3: 45.
https://doi.org/10.3390/fractalfract3030045