If $f: X \rightarrow Y$ is $C^0$ with $X$ and $Y$ arcwise connected, then $f_{*}$ is isomorphism to homology.

Question

If $f: X \rightarrow Y$ is $C^0$ with $X$ and $Y$ arcwise connected, then $f_{*}:H_0(X) \rightarrow H_0(Y)$ is isomorphism.

This is exercise of algebraic topology. I know that $f_{*}$ is homomorphism but I don't see why is bijectiv. Thank you for you help!

Note: Since $Y$ is arcwise connected, I know that $H_0(Y)= \mathbb{Z}$.