Are there continuous functions for which the epsilon-delta property doesn't hold? [duplicate]

The statement you made is the definition of continuity (in a metric space, where the notion of distance or $|x-x_0|$ makes sense). So it works both ways: If the function is continuous, it meets the definition and if it meets the definition then it is continuous.

You are encountering the idea of a function being "uniformly continuous." In the definition you wrote, there is nothing said about whether then same $\delta(\epsilon)$ can work irrespective of the point $x_0$. Your exponential example is indeed continuous, but it is not uniformly continuous, because $\delta$ has to depend both on $\epsilon$ and $x_0$.

In contrast, a function like $\frac{x^3}{1+x^2}$ is uniformly continuous, because for any given $\epsilon > 0$ you can find a $\delta$ such that that defining condition holds for every real $x_0$.

Since it's a definition, your first "if" should really be read as an "if and only if". So the converse you ask about always holds.