Can the Principle of Mathematical induction be generalized?

Question

For example, if we have a statement $f(x)$ which we have to prove whether it is true for all $x$ in the set $(0,\infty)$. Now, we first prove that $f(0)$ is true. Then, we assume $f(k)$ is true. Then we prove that $\lim_{h\rightarrow 0}f(k+h)$ true. Now, this doesn't look good to me because $\lim_{h\rightarrow 0}f(k+h)$ is exactly equal to $f(k)$ which we've assumed to be true. But, can something be done to prove a statement for all reals?