I don't understand the following:
Let $(X,d_x)$ and $(Y,d_y)$ be metric spaces, and let $f:X\to Y$ be a function.
Definition (of continuous function)
A function $f$ is called continuous at $x\in X$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that $d_y(f(x),f(y))<\varepsilon$ whenever $d(y,x) < δ$.
Definition (of limit of function)
..
for every $\varepsilon > 0$ there exists $\delta > 0$ such that $d_y(f(x),l)<\varepsilon$ whenever $d(y,x) < δ$ $y\neq x$.
I don't really understand why we say that $y\neq x$ in the definition of limit of function and why we don't use $y\neq x$ in the definition of continuous function.
Thanks for your help!
Best Answer
Continuity at $x$ is stronger than the existence of a limit at $x$.
Specifically, $f$ is continuous at $x$ if and only if
The first bullet point explains why "$l$" in the definition of limit is replaced by "$f(x)$" in the definition of continuity . The second bullet point explains why you need to remove the $y \ne x$ restriction.