Question

prove that if f and g are continuous at x=a, then f+g is continuous at x=a

Answer 1

if function is continuous, it means that for small change in x the change in y is also small. Writen more presisly:
\[for \,\epsilon>0 \, \exists \, \delta >0 \, : if\, |x-x_{0}|<\delta \rightarrow \, |f(x)-f(x_{0}||<\epsilon\]
We can use this to prove your statement.
Since bouth , f and g are continous at x=a, it means that for bouth there exist their own delta which makes the change in f or g less than any chousen epsilon_f and epsilon_g. So for f+g, chousen any arbitrary small number, let's call it epsilon_a, we can choose a delta that would make epsilon_f+epsilon_g<epsilon_a. Which orves the continuiny of f+g

Answer 2

Can the same argument be applied to proving the statement for f-g?

Answer 3

yes