$8$. (b) Let $Y$ be an ordered set in the order toppology. $f,g:X\rightarrow Y$ be continuous. Define the function $h:X\rightarrow Y$ by $$h(x)=\min\{f(x),g(x)\}$$ Prove that $h$ is continuous. [Hint: Use the Pasting Lemma]
To show that a function is continuous, we have to show that for any open set $V\subseteq Y$, the preimage $h^{-1}(V)$ is open. How would we show this? A concrete example would really help. I don't need the full solution.
$h(x)=f(x)$ on $\{x|f(x)≤g(x)\}$ , and $h(x)=g(x)$ on $\{x|f(x)\ge g(x)\}$ , therefore, $h$ restricted to these sets is continuous, and both sets are closed. Using the pasting lemma, $h$ is continuous.