A dumb question on continuity and differentiability of function

The derivative $f'(0)$ is defined by

$$\lim_{h\to 0} \frac{f(0+h)- f(0)}{h}$$

so you are evaluating $f$ at $x$-values other than $c$. So the question comes down to whether the values of $\frac{f(0+h)-f(0)}{h}$ converge to a value (from both sides) around $0$.

Differentiating $c$ is therefore a non-starter (as you've realised); differentiating $\frac{x}{e^x -1}$ is on the right track but you shouldn't just be thinking in terms of substituting $0$ into that derivative.