Q:

Epsilon delta with point discontinuity?

Please forgive me if this question has a trivial answer, but I’m a bit stuck after coming up with this scenario. If there is a function with a point/removable discontinuity, for example $f(x)=1$ when $x\neq 0$ but $f(x)=0$ when $x=0$. The limit as $x$ approaches $0$ is $1$ but with the epsilon delta definition, if we take $\epsilon = 0.1$, then if $x=0$ (which is in the range $|x-a|<\delta$), $|f(x)-1|>0.1$, which is the negation of the epsilon delta statement. What exactly am I missing?

A:

The $\epsilon-\delta$ definition for continuity is different form the $\epsilon-\delta$ definition for limit, notably for continuity the value $x=x_0$ is allowed but not for the limit for which $x\neq x_0$.

