Composed interval/Continuous function to metric space/Exercise

Let ${}(M,d)$ be a metric space, and let ${}a<b<c$ be real numbers. Let

$f\colon [a,b]\longrightarrow M$

and

$g\colon [b,c]\longrightarrow M$

be continuous mappings with ${}f(b)=g(b)$. Show that the mapping

$h\colon [a,c]\longrightarrow M$

given by

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

is also continuous.

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