# Replace the epsilon and delta in the definition of continuity, what kind of continuity do we get?

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?

• 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 $x_0.$ – Dbchatto67 Mar 21 at 6:57
• The function is bounded in the second case – vidyarthi Mar 21 at 7:24
• @Dbchatto67 Due to my humble love of math – High GPA Mar 21 at 7:27
• why the downvote? – vidyarthi Mar 21 at 7:28
• see here – vidyarthi Mar 21 at 7:30