Restriction of a Continuous Map, and Continuous Images of Compact Subsets

Let $(X,\mathcal{T}_{X})$ and $(Y,\mathcal{T}_{Y})$ be topological spaces, let $f:X\to Y$ be a continuous map, and let $A\subseteq X$ be equipped with the subspace topology $\mathcal{T}_{A}$. Let $f|_{A}:A\to Y$ denote the map with $f|_{A}(x)=f(x)$ for every $x\in A$, and write $f(A)=\{f(x):x\in A\}$. Then the following hold.

1. (Restriction) The map $f|_{A}$ is continuous from $(A,\mathcal{T}_{A})$ to $(Y,\mathcal{T}_{Y})$.

2. (Image of a compact subset) If $A$ is compact in $X$, then $f(A)$ is compact in $Y$.