Union of intervals/Continuous function to R/Exercise

Let ${}a<b<c$ be real numbers and let

$f\colon [a,b]\longrightarrow \mathbb {R}$

and

$g\colon [b,c]\longrightarrow \mathbb {R}$

be continuous functions such that ${}f(b)=g(b)$. Prove that the function

$h\colon [a,c]\longrightarrow \mathbb {R}$

such that

$h(t)=f(t){\text{ for }}t\leq b{\text{ and }}h(t)=g(t){\text{ for }}t>b$

is also continuous.

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