Let a be less than b which is less than c.
Suppose that f is continuous on [a,b], that g is continuous on [b,c], and that 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]
Let a be less than b which is less than c.
Suppose that f is continuous on [a,b], that g is continuous on [b,c], and that 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]
The only point in question is $\displaystyle b$. Since $\displaystyle f$ is continuous, $\displaystyle \lim_{x\to b^-}h(x)=\lim_{x\to b^-}f(x)=f(b)=h(b)$
Since g is continuous, $\displaystyle \lim_{x\to b^+}h(x)=\lim_{x\to b^+}g(x)=g(b)=f(b)=h(b)$
So $\displaystyle \lim_{x\to b^-}h(x)=h(b)=\lim_{x\to b^+}h(x)$, and $\displaystyle h$ is continuous at $\displaystyle b$ and therefore on $\displaystyle [a,c]$.