Properties of Continuous Functions

If two functions $f$ and $g$ are each continuous at $x=a$, then so are the following combinations, built from them:

1. their sum $(f+g)$ is continuous at $x=a$;
2. their difference $(f-g)$ or $(g-f)$ is continuous at $x=a$;
3. the constant multiple $k\cdot f$, for any $k\in\mathbb{R}$, is continuous at $x=a$;
4. their product $(f\cdot g)$ is continuous at $x=a$;
5. their quotient $\dfrac{f}{g}$, provided $g(a)\ne0$, is continuous at $x=a$;
6. their composite function, $f[g(x)]$ or $g[f(x)]$ — that is, $f\circ g(x)$ or $g\circ f(x)$ — is continuous at $x=a$. …