TITLE: Example of topological spaces where sequential continuity does not imply continuity
QUESTION [12 upvotes]: Please give an example of a function $f : X \to Y  $ where $X,Y$ are topological space , such that there exist $x \in X$ such that for every sequence $\{x_n\}$ in $X$ converging to $x$ , $\{f(x_n)\}$ converges to $f(x)$ but $f$ is not continuous at $x$ ; also please give such an example that $f$ is not continuous any where in the domain but for every $x \in X$ and sequence $\{x_n\}$ in $X$ converging to $x$ , $\{f(x_n)\}$ converges to $f(x)$.

REPLY [2 votes]: Let $\omega_1$ be the first uncountable ordinal and take $X=[0, \omega_1]$ (so really $\omega_1\cup\{\omega_1\}$) with the order topology. Every sequence converging in $[0,\omega_1)$ is bounded, hence every sequence in $X$ which converges to $\omega_1$ must be eventually constant. Therefore any $f\colon X\to Y$ is sequentially continuous in $\omega_1$. Now you set take $f(x)=0$, if $x<\omega_1$ and $f(\omega_1)=1$. This function is not continuous in $\omega_1$, because $\{\omega_1\}$ is not open, since $\omega_1$ is not a successor.