Normally continuity of a map between normed spaces is defined as :
A map $f : X \rightarrow Y$ is continuous at $x_0$, if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x \in X$ satisfying $||x - x_0\|| < \delta $, we have $||f(x) - f(x_0)|| < \epsilon$.
I am wondering that if the following is equivalent:
A map $f : X \rightarrow Y$ is continuous at $x_0$, if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x \in X$ satisfying $||x - x_0\|| \leq \delta $, we have $||f(x) - f(x_0)|| \leq \epsilon$.
Perhaps I am being overly analytic, but I would still like to know if there is any harm is using the latter definition.