Take $f(x)$ and $g(x)$ and assume, first, that they are both continuous. Then using the fact that each of $f(x)$ and $g(x)$ is continuous (strictly defined), then establish that $h(x) = f(x) + g(x)$ (or $h(x) = f(x) - g(x)$) must also be continuous.
Put differently: What is required for $h(x)$ to be continuous? Establish that we can satisfy this requirement (definition) provided $f(x)$ is continuous, by definition, and likewise $g(x)$ is continuous.
Do the same with $f(x)$ and $g(x)$ being differentiable functions. What can we then conclude about the differentiability of $h(x) = f(x) \pm g(x)$?
You'll see that your intuition can readily be justified.
- Recall: If $\lim_{x\to a}f(x)$ and $\lim_{x\to a}g(x)$ both exist, then if $h(x) = f(x) + g(x)$, we have that $\lim_{x\to a}h(x)$ exists and $\lim_{x\to a}h(x) = \lim_{x\to a}(f(x) + g(x)) = \lim_{x\to a}f(x) + \lim_{x\to a}g(x).$
- The above can be proven using the standard $\epsilon-\delta$ definition.
On a side note: I don't know of any serious math student who *doesn'*t, at some point along the way, question the certainty of what one s\he learned. That's to be expected, and once you get on firm ground again, which you will, you'll likely encounter that feeling again, down the road. It's actually a good sign that you are learning, and are in the process of acquiring an understanding of what you've previously learned at a far deeper level than ever before...