$(1) \Rightarrow (2)$ The Pasting Lemma states that if $f_1 : X_1 \rightarrow Y$ and $f_2 : X_2 \rightarrow Y$ where $X_1$ and $X_2$ are closed subset of $X$ and $f_1, f_2$ are continuous, and the two functions agree on the overlap, then $f$ defined by "pasting" the two function together is continuous. See page 108 in Munkres $\textit{Topology}$.
$(2) \Rightarrow (1)$ If $f$ is continuous, then for all $U \subset Y$ which is open, $f_1^{-1}(U) = f^{-1}(U) \cap X_1$. $f^{-1}(U)$ is open by continuity of $f$. $f^{-1}(U) \cap X_1$ is open in the relative topology of $X_1$ by definition. So $f_1$ is continuous on $X_1$ with the relative topology. Same for $X_2$.
I think there is probably a more elementary proof of the first direction rather than using the pasting lemma. However, the pasting lemma is not hard and quite useful.
In Response to the comment I will clarify the notion of a restriction.
If $f : X \rightarrow Y$ is a function between topological spaces and $X_1 \subset X$ ,then $f_1 : X_1 \rightarrow Y$ defined by $f_1(x) = f(x)$ for all $x \in X_1$ is a function between the topological space $X_1$ to $Y$ where the topology on $X_1$ is the subset or relative topology. That is $U \subset X_1$ is a open if and only if there is a $V$ open in $X$ such that $U = V \cap X_1$.