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Show if $f$ is continuous in $x \iff$ for all $\epsilon>0$ there exists some $\delta >0$ such that for all $\zeta \in (a,b): |f(\zeta) - f(x)| < \epsilon$, whenever $|\zeta - x| < \delta$.

I know if $f$ is continuous in $x$, then $\lim_{\zeta\to x} f(\zeta) = f(x)$.

But how can I use that to show equivalence for both statements?

RobPratt
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skinnyBug
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    What does $\lim_{\zeta\to x}f(\zeta)=f(x)$ mean? –  Nov 29 '21 at 19:48
  • sry did forget sth..... – skinnyBug Nov 29 '21 at 19:50
  • @SaucyO'Path Peraphs the user thought a generalization of a continuous function writing $x$ like a fixed point. – Sebastiano Nov 29 '21 at 19:56
  • @ArminZierlinger Please write the definition of a limit in the same way that you wrote the definition of continuity – Ben Grossmann Nov 29 '21 at 19:56
  • @Sebastiano Excuse me, what? –  Nov 29 '21 at 19:56
  • @SaucyO'Path I have thought that the function is $y=f(\zeta)$ and $f(x)$ it is the computation of $f(\zeta)$ in $x$. – Sebastiano Nov 29 '21 at 19:58
  • Related duplicate? https://math.stackexchange.com/questions/494767/show-continuity-using-epsilon-delta-definition or https://math.stackexchange.com/questions/600788/epsilon-delta-definition-to-prove-that-f-is-a-continuous-function – Sebastiano Nov 29 '21 at 19:59
  • Hint: Can you write exactly the definition of convergence, i.e. the signification of $\lim_{\zeta\to x} f(\zeta) = f(x)$ ? – Leo Nov 29 '21 at 22:22

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