Task from exam: With $\varepsilon-\delta$ define what it means: $f$ is continuous in $x_{0}$

Question

I found an old exam and one task is: What does "$f$ is continuous in $x_{0}$" mean, with reference to $\varepsilon-\delta$ definition.

$f$ is a function and $x_{0}$ is the position where we check if the function is continuous...

Would it be enough if I just wrote:

Let $f: A \rightarrow \mathbb{R}$

$f$ is continuous in $x_{0}$ $\Leftrightarrow$ for all $\varepsilon>0 $ there exists a $\delta>0$ so that for all $x \in A:$ $|x-x_{0}|<\delta: |f(x)-f(x_{0})|< \varepsilon$

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Is it correct like that? Or would you do it completely different than me? Please let me know, it's very important!

Answer 1

Your definition is correct if $A$ is a subset of $\mathbb R$. Otherwise, $|x-x_0|$ is not defined.

Also, it would be nice (but it's not really a mistake if you don't) to mention somewhere that $x_0\in A$.

Answer 2

Your answer is wrong due to the use of colons. A colon means "such that", which means the last colon is wrong. A correct definition would be:

for all $\varepsilon>0 $ there exists a $\delta>$ so that for all $x \in A:$ $|x-x_{0}|<\delta \Rightarrow |f(x)-f(x_{0})|< \varepsilon$