Differentiability versus Continuity

Two students of calculus argue over whether a continuous function $f(x)$ with no removable discontinuities or singularities over an interval $[a,b]$ must have a derivative $f'(x)$ .

What do you think?

note: A removable discontinuity at $x=c$ means the limit exists there, but the functional value does not, or is not the same as the limit.

A singularity at $x=c$ means the limit does not exist and the function can tend towards $\pm\infty$ . The function value at $x=c$ may or may not be well defined.