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Let $F=\{f: \mathbb{R} \to\mathbb{R}:|f(x)-f(y)|\leq K|(x-y)^\alpha|\}$ for all $x,y\in\mathbb{R}$ and for some $\alpha >0$ and some $K > 0$ .
Which of the following is/are true?
1. Every $ f \in F$ is continuous
2. Every $f\in F$ is uniformly continuous
3. Every differentiable function $ f$ is in $F$
4. Every $f \in F$ is differentiable

The given condition is Lipchitz condition. So 1 and 2 are true. but what can I say about the others

  • 4
    The given condition is Holder continuous which is not the same as uniform continuity.2012-12-17
  • 0
    Is every differentiable function uniformly continuous?2013-05-29
  • 0
    There is a bit of vocabulary-based confusion here. These days, we tend to call this condition Holder continuity. However, in the older literature (e.g. harmonic analysis papers from the 1970s), this is often called a Lipschitz condition with an exponent.2013-05-29

2 Answers 2