Is this set countable or uncountable? (Related to mean value theorem) Let $f$ be a differentiable function on the real line.

Consider any $h\in (0,1)$. By the mean value theorem there exists $d_h\in (0,h)$ such that $f'(d_h)=\frac{f(h)-f(0)}{h}$.
Is the set $\{d_h\}$ countable or uncountable?

(Update: I wish $h$ to be varied, not fixed. For every new value of $h$, there is this new $d_h$ indexed by $h$ produced by the mean value theorem. So the question is that is the set of all the $d_h$ countable or not.)

There are certainly uncountable choices of $h$, but there may be repetitions such that $d_{h_1}=d_{h_2}$.

Thanks.
Let $g:(0,1)\rightarrow\mathbb{R}$ be defined by $g(h) = \frac{f(h)-f(0)}{h}$. Then $g$ is continuous. If $g$ is not constant, then by the intermediate value theorem its range must be uncountable, and hence the set of $d\in(0,1)$ such that $f'(d) = g(h)$ for some $d&lt;h&lt;1$ must also be uncountable. If $g$ is constant, then $f$ is linear, and so $f'(d) = g(h)$ for any $d,h\in(0,1)$, i.e. the desired set is also uncountable.
If $f$ is constant then the set is obviously uncountable.

If $f$ is linear that is $f(x)= ax +b $ then the set is obviously uncountable.
If we treat $h$ as a parameter for non-constant  $f$ and let it vary then the set $\{d_h \}$ is uncountable. 
If $h$ is fixed for non-constant and non-linear analytic $f$ the set is countable.