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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.

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    What you want to show is false; $h$ will be continuous at $b$. For points in $[a,c]$ that are actually in $[a,b)$, you can use the continuity of $f$ directly, taking care not to "go" past $b$; for points that are in $(b,c]$, you can use the continuity of $g$ (again, taking care not to go "past" $b$); for the point $b$, you'll need to combine the fact that $f$ is continuous at $b$ (remember that this means that $f$ is continuous **from the left** at $b$) and that $g$ is continuous at $b$ from the right.2012-03-25

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