Does the function $f: \mathbb{R} \rightarrow \mathbb{R}$ exist if
- $f(1) = 1$ and $f(-1) = -1$, and
- $|f(x) - f(y)| \leq |x-y|^{3/2}$ for all $x,y \in \mathbb{R}$.
To decide if the function exists or does not exist?
0
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real-analysis
functions
1 Answers
2
Hint:
\begin{align} \left|\frac{f(x)-f(y)}{x-y}-0\right| \leq |x-y|^{1/2} \end{align} which means $f$ is differentiable. So what is the derivative at every point?