Is there any differences between a continuously differentiable function and a common function? I want ask this question because I have seen many exercises telling me that f(x) is continuously differentiable.
For example, suppose f(x) is continuously differentiable, and f(0)=0 $$f'(x)+\int_0^x f(t)\,\mathrm{d}t=e^x$$ Find out the expression of f(x).
Why do I need f(x) to be continuously differentiable?
The equation you have quoted does not make sense if $f$ is not differentiable. If $f$ is differentiable and the integrable (so that the RHS makes sense) then automatically $f'(x)=e^{x}-\int_0^{x} f(t) \, dt$ is continuous. Hence there is no hope of any solution to the equation unless $f$ is continuously diffrentiable.