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Suppose that $f$ is a continuous function on $\mathbb{R}$. I want to estimate the definite integral $$ I:= \int_{0}^a [f(x)-f(0)]dx $$ by the upper bound $M = \sup_{x\in[0,a]}|f(x)|$ and the variation …

Suppose that $f$ is a continuous, nonconstant function on $[0,1]$. Fix some $0<a<1$. Is it possible to establish the following inequality $$ |f(x+h)-f(x)| \leq C \left[ |h|^a + |2f(x)-f(x+h)-f(x-h)| \ …

Suppose that $f$ satisfies $a$-Hölder condition on $[0,1]$ ($0<a<1$). Fix $0<b<a$. For any $x\in [0,1]$ and sufficiently small $\delta>0$, is the following inequality established? $$ \int_0^\delta t^{ …

Fix a continuously differentiable but nowhere twice differentiable function $f$ on $\mathbb{R}$ supported on $[0,1]$. Is it true that for all $x\in[0,1]$ and all $\delta$ sufficiently small \begin{ali …

Suppose that $f$ is continuous on $[0,1]$. Thus, $f\in L^1(\mathbb{R})$ and its Fourier transform exists, as $$ \hat{f}(\xi) := \int_\mathbb{R} e^{-2\pi i x \xi} f(x)dx, $$ which can also be written a …

Let $f$ be a bounded and continuous function, $0<a < 1$. $U(x,r)$ is the neighborhood of $x$ with diameter $r$. Can we prove the following equation of two limits $$ \lim_{r\rightarrow 0} \sup_{y,z \i …

It is known that if a real continuous function $f(x)$ satisfies a local $\alpha$-Hölder condition on a closed interval $[a,b]$, the box dimension of the graph of $f(x)$ on $[a,b]$ will be not greater …

Let $f$ be a function on some real interval $[a,b]$. Suppose that $\forall x\in [a,b]$, there exists a positive constant $C$ such that $$ |f(x)-f(y)| \leq C|x-y| $$ for all $y \in [a,b]$. Does each $x …

Suppose that $f(x)$ is continuous on $[0,1]$. We make an agreement that if there exists an interval $[a,b]\subseteq[0,1]$ including point $y$ such that $f(x)$ satisfies $\alpha$-Holder condition on $[ …