Did you consider looking up "continuous" in your text?
That property pretty much is the definition of "continuous":
The function f(x) is continuous at x= a if and only if
(1) f(a) exists.
(2) [itex]\lim_{x\to a} f(x)[/itex] exists.
(3) [itex]\lim_{x\to a} f(x)= f(a)[/itex].
Since (3) pretty much implies the left and right sides exist, of only that is given as the definition- but that's really "shorthand".
well, I meant that the function needs to be continuous at the point x=c. There's a theorem in Calculus that talks about the composition of functions. the theorem states that if [itex]lim_{x→b}f(x)=c[/itex] and [itex]lim_{x→a}g(x)=b[/itex] then [itex]lim_{x→a} fog(x)=c[/itex] is NOT true in general, but this law holds if f is continuous at x=b.
That's why I said continuity is the key. The theorem seems to be easy to be proved though.