Suppose $f: ℝ \rightarrow ℝ$ is differentiable. Show that if $c \in ℝ$ and $\lim_{x \rightarrow c} f'(x)$ exists, then $$f'(c) = \lim_{x \rightarrow c} f'(x)$$
The hint on this question is that derivatives have the Intermediate Value Property. Other than that, I really don't know where to start with this.
Once I get this started, I'm sure I'd be able to figure it out from there.
Best Answer
$\dfrac {f(x)-f(c)}{x-c}=f'(t_x)$, where
$t_x \in (\min (c,x), \max (c,x))$.
$\lim x \rightarrow c$ implies $\lim t_x \rightarrow c$.
$\lim_{x \rightarrow c} \dfrac{f(x)-f(c)}{x-c}=$
$\lim_{t_x \rightarrow c} f'(t_x) =L.$
$ \lim_{ x \rightarrow c} \dfrac{f(x)-f(c)}{x-c}$ exists and is equal to $L,$ $f$ is differentiable at $c.$