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Let $f:R \rightarrow R$ is a real valued function $\forall$ $x,y \in \mathbb{R}$ such that $$|f(x)-f(y)| \leq |x-y|^3$$

Prove that $h(x) = \int f(x)dx$ is continuous $\forall$ $x \in \mathbb{R}$

Now, my thought process here is I'll probably need to procced by using differentiability and differentiating the function, since I can see it having the form of differentiation by first principle when I just rearrange the equation a bit: $$\frac{|f(x)-f(y)|}{|x-y|} \leq |x-y|^2$$

But I dont know how to proceed here? Also, what should be my thought process approaching this question here? I know that proving that a function is differentiable also proves that it is continuous, but how do I go about differentiating here after I get the above form?

Techie5879
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1 Answers1

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Take limit in your inequality as $y \to x$. You see that $f'(x)$ exists and equals $0$. Since this is true for all $x$, $f$ must be a constant.

Kavi Rama Murthy
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