Calculus of Continuity
Theorem: CCD.CC: Suppose $\lim_{x \to a}f(x)= f(a)$ , $\lim_{x \to a}g(x)= g(a)$,  and $\alpha$ is a real number,then
(i) $\lim_{x \to a} \alpha f(x)= \alpha f(a)$;
(ii) $\lim_{x \to a} f(x) \pm g(x) = f(a) \pm g(a)$;
(iii) $\lim_{x \to a} f(x) \cdot g(x) = f(a) \cdot g(a)$;
(iv) If $M \ne 0$,  then $\lim_{x \to a} \frac {f(x)}{g(x)} = \frac {f(a)}{g(a)}$;
(v) If $\lim_{y \to f(a)} g(y)= g(f(a))$, then $\lim_{x \to a}g(f(x))= g(f(a))$.

Discussion and Some Proofs of CCD.CC