Is the continuous image of a complete metric space complete?

Question

Let $(X,d)$ be a complete metric space and $f:X\longrightarrow X$ be a continuous mapping. Then is it true that $f(X)$ is complete?

I can't seem to get the result in the affirmative unless $d(x,y)\leq d(f(x),f(y))$.

Any help would be appreciated. Thanks in advance!

Answer 1

No, unless we assume some more conditions on $X$ or $f$.