Is it possible to find that $f(0) = 1$ with the only hypothesis $f(x) = f'(x)$?

Question

I often read in maths books when attemting to find to find the function which has the property $f(x) = f'(x)$, that by assumption $f(0) = 1$.

Is there a way to find this result $f(0) = 1$, with the only hypothesis that $f(x) = f'(x)$ ?

Thank you for your concern.