In his book Differentiable manifolds, Boothbay says

If $f:U\rightarrow\mathbb{R}$ is differentiable at every point of open set $U$ in $\mathbb{R}^n$ then we say that $f$ is differentiable on $U$. Note that differentiability is a local concept, i.e. if $f$ is differentiable on a neighbourhood of each point of $U$ then $f$ is differentiable on $U$. By the mean value theorem, for a function of one variable the existence of the derivative at $a\in U$ is equivalent to differentiability.

I could not understand the last line By the mean value theorem ....

The mean value theorem I had never seen implies local property of differentiability.

Can one explain the last statement in simple way?

  • 2
    $\begingroup$ No need for the mean value theorem, simply a directional derivative in dimension $1$ is the derivative (because there are not further directions). $\endgroup$ – John B May 19 '18 at 11:05

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