To prove continuity of compositions of functions, a helpful rule is: if f and g are continuous (at "matching" points or on "matching" intervals), so is the composition g∘f (think: "apply f first, then g"). More precisely, if f(x) is continuous at x=a and g(y) is continuous at the corresponding point y=f(a), then (g∘f)(x) = g(f(x)) is continuous at x=a. Similarly, if f(x) is continuous on an interval I and g(y) is continuous on the corresponding interval f(I), then (g∘f)(x) = g(f(x)) is continuous on the interval I.