complete subset of a metric space

Question

Let $f:X\to Y$ be a continuous map between metric spaces. Then $f(X)$ is a complete subset of $Y$ if

A. the space $X$ is compact

B. the space $Y$ is compact

C. the space $X$ is complete

D. the space $Y$ is complete

I am unable to arrive at a conclusion.

Answer 1

The answer is $A$.

Since $f$ is continuous and $X$ is compact, $f(X)$ is compact. A compact subset of a metric space is complete, so $f(X)$ is complete.