EXAMPLE 9 Video Example Where are the following functions continuous? (a) h(x) = sin(x³) (b) F(x) = In(1+ cos(x)) SOLUTION (a) We have h(x) = f(g(x)), where g(x)=x² and f(x) = X We know that g is continuous on R since it is a polynomial and f is also continuous everywhere. Thus, fog is continuous on R by this theorem. (b) We know from this theorem that f(x) = In(x) is continuous and g(x) = 1 + cos(x) is continuous (because both y = 1 and y = cos(x) are continuous). Therefore, by this theorem, F(x) = f(g(x)) is continuous wherever it is defined. The expression In(1 + cos(x)) is defined when 1 + cos(x) > . So it is undefined when cos(x) = Thus, F has discontinuities when x is an odd multiple of it and is continuous on the intervals between these values. (See the figure below.) 10 5 mdr. T -10 L -10 10 X G , and this happens when x ±, ±3m,....