If the Continuous image of a space is compact, does that mean the space is compact?

Question

Let $f$ be a continuous function and $f(X)$ is compact. Is $X$ necessarily compact?

Is there an example to prove/disprove this?

Thank you.

Answer 1

Let be $X$ any space and $f$ constant...

Answer 2

The image of $(-1,1]$ by $x \to x^2$ is compact.