Exercise 8.2.16

Question

Show that if $f \in C[0,1]$ is differentiable at a point $x \in [0,1]$, then $f \in A_{m,n}$ for some pair $m, n \in \mathbf{N}$.

Ulisse Mini · Accepted Answer

Let $f$ be differentiable at $x$, and let $\epsilon = 1$. Then there exists $\delta > 0$ such that $|x - t| < \delta$, $$ \left|\frac{f(x) - f(t)}{x - t}\right| - \left|f'(x)\right| \leq \left|\frac{f(x) - f(t)}{x - t} - f'(x)\right| < \epsilon = 1$$ So, for $m > 1/\delta$ and $n > f'(x) + 1$, $f \in A_{m,n}$.

Exercise 8.2.16

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Exercise 8.2.16

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