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1) $f : X \to Y$ is a map. For all $x_0\in X$, if for every $\delta > 0$, there is $\varepsilon > 0$ such that $d_X(x_0, x) < \delta$ whenever $d_Y(f(x_0), f(x)) < \varepsilon$.

2)$f : X \to Y$ is a map. For all $x_0\in X$, if for every $ \varepsilon> 0$, there is $\delta > 0$ such that $d_Y(f(x_0), f(x)) < \delta$ whenever $d_X(x_0, x) < \varepsilon$.

Is map $f$ continuous in the two cases? If not, what kind of continuity is this?

For the second the case, I guess this definition is much stronger than continuity?

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    $\begingroup$ You have shown that $f^{-1}$ is continuous at $f(x_0)$ (if it exists at all) instead of showing that $f$ is continuous at $x_0.$ $\endgroup$ – Dbchatto67 Mar 21 at 6:57
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    $\begingroup$ The function is bounded in the second case $\endgroup$ – vidyarthi Mar 21 at 7:24
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    $\begingroup$ @Dbchatto67 Due to my humble love of math $\endgroup$ – High GPA Mar 21 at 7:27
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    $\begingroup$ why the downvote? $\endgroup$ – vidyarthi Mar 21 at 7:28
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    $\begingroup$ see here $\endgroup$ – vidyarthi Mar 21 at 7:30

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