Transfer of Ideas Math Example 4

Solution

1. 1
   Assume the hypothesis: $f$ is continuous at $a$ (i.e., $\lim_{x\to a}f(x)=f(a)$) and $g$ is continuous at $a$.
2. 2
   Derive: $\lim_{x\to a}(f+g)(x) = \lim_{x\to a}f(x)+\lim_{x\to a}g(x) = f(a)+g(a) = (f+g)(a)$.
3. 3
   Conclusion: $f+g$ is continuous at $a$.