Proving the continuity of the exponential function

Question

Given the series definition of the exponential function, i.e. $\exp(x) = \displaystyle\sum\limits_{k = 0}^{\infty} \dfrac{x^k}{k!}$. Given that I have already proven that polynomials are continuous, does from this fact follow the continuity of the exponential function ?

Answer 1

Almost. The fact that polynomial functions are continuous together with the fact that that power series converges uniformly to the exponential function on any bounded subset of $\Bbb R$ is enough, since uniform convergence preserves continuity.