Let S (subset) R, and let f, g be functions from S to R. If f and g are continuous at
a ia an element of set S, then prove that h(x) = f(x) + g(x) is also continuous at a.
Hint: look at the proof of the arithmetic of sequences.
A function is continuous at point a if
$\displaystyle \lim_{x\to{a}} f(x) = f(a) $ AND f(a) exist.
If you already know that the two functions are continuous at a then
$\displaystyle \lim_{x\to{a}} h(x) = \lim_{x\to{a}} f(x)+g(x) = \lim_{x\to{a}} f(x) +\lim_{x\to{a}} g(x) = f(a) + g(a) = h(a)$
All this by definitions.