I am trying to rephrase the definition of continuity to see if I have the right intuition. Please comment on how I might be wrong or misunderstanding something:
A function f is continuous at a point $x=a$ if when you fix $x=a$ and fix $\epsilon$ there is a delta neighborhood around $x=a$ which will ensure that any $x$ in that neighborhood $\neq$ a will ensure an $f(x)$ that is in the $\epsilon$ neighborhood of $f(a)$. But if you keep the same $\epsilon$ and fix another $x=b$, the delta neighborhood you need around that new $x=b$ may not be the same as the first one to ensure that all the members of the neighborhood ensure an $f(x)$ in the $\epsilon$ neighborhood of $f(b)$