Theorems of Continuity

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A function defined in a closed interval [a,b] is continuous in [a,b] if and only if it is continuous in the open interval (a,b), as well as continuous from the right at "a" and from the left at "b"

If f and g are continuous functions at a, then so are the functions f + g, f - g, fg, and f/g where g(a) does not = 0.

If lim g(x) = b and f is continuous at b, lim f(g(x)) = f(b) = f[lim g(x)] while all limits are x--> a.

If g is continuous at a and f is continuous at b = g(a), then lim f(g(x)) = f[lim g(x)] = f(g(a)) while all limits have x as going toward a.

Intermediate value Theorem. If f is continuous on a closed interval [a,b] and if f(a) does not = f(b), then f takes on every value between f(a) and f(b) in the interval [a,b].