Difference between limit of function and continuous function in metric spaces

metric-spaces

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 limit of $f$ at $x$ exists and equals $l$, and
  • $l = f(x)$.

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.

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