This is a question from my math book:
Let $a< b < c$. Suppose that $f$ is continuous on $[a,b]$, $g$ is continuous on $[b,c]$, and $f(b) = g(b)$. Define $h$ on $[a,c]$ by $h(x) = f(x)$ for $x\in [a,b]$ and $h(x) = g(x)$ for $x \in [b,c]$. Prove that $h$ is continuous on $[a,c]$.
What I want to do is prove that $h$ is continuous on $[a,c]$ but not at $b$.
I'm thinking that I have to pick an $α$ such that $a<α< b$, and then show that $h(x)$ is continuous at $α$. So basically, I want to show that $∀\epsilon>0, ∃ δ>0$ such that if $x\in [a,c]$ and $|x-α|<δ$ then $|h(x)-h(α)|<\epsilon$. And from the given information, I know that $f:[a,b]\to \mathbb{R}$, and $∀ \epsilon>0, ∃ δ>0$ such that if $x\in [a,b]$ and $|x-α|<δ_f$ then $|f(x)-f(α)|<\epsilon$.
How do I connect these two definitions to find $δ$? And is there anything else I need to prove? Like are there any cases I should be making?
Thanks in advance.