$x_{n} \rightarrow x$ and $f(x_{n}) \rightarrow f(x)$ is $f$ continuous?

Question

Given a metric space $(M,d),$ let $f:M\to \mathbb R$ be a function such that for all convergent sequences $x_n\to x$ in $M$, we have $f(x_n)\to f(x)$ in $\mathbb R.$ Is $f$ continuous?

How can I prove or disprove ?

This is a doubt which I had! I am sorry but my doubt didn't necessarily have any other attributes associated with the sequence.

Answer 1

Assume $f$ is not continuous at $x.$

Then there is a $\epsilon>0$ such that for all $\delta>0,$ there is a $y_\delta$ such that $d(x,y_\delta)<\delta$ and $|f(x)-f(y_\delta)|\geq \epsilon.$

Now, consider $x_n=y_{1/n}.$ Show that $x_n\to x$ but $f(x_n)$ does not converge to $f(x).$