continuity of a composed function

Question

Let $X$ be a topological space and $H_1:[0,0.5] \to X$ and $H_2:[0.5,1] \to X$ be continuous with $H_1(0.5)=H_2(0.5)$. Why is $H(x) :=\left\{\begin{array}{ll} H_1(x), & x \leq 0.5\\ H_2(x), & x \geq 0.5\end{array}\right.$ continuous on $[0,1]$?

Answer 1

This is a special case of the pasting lemma.

If $X=A\cup B$ with both $A,B$ closed in $X$, $f:A\to C$ and $g:B\to C$ are both continuous and they agree on $A\cap B$, then their "glueing"

$$h:X\to C$$ $$h(x)=\begin{cases} f(x) &\text{when }x\in A \\ g(x) &\text{when }x\in B \end{cases}$$

is well defined and continuous as well.

Indeed, in that situation if $F\subseteq C$ is closed, then $h^{-1}(F)=f^{-1}(F)\cup g^{-1}(F)$ which are both closed subsets of $A$ and $B$ respectively. And thus they are closed in $X$ and so is their union.