True or False? Convergence

Question

True/ False: If f is continuous on [0, ∞) and the (integral of 1 to ∞) from f(x)dx is convergent, then the (integral of 0 to ∞) from f(x)dx is convergent

Answer 1

This is a true statement. Since f(x) is continuous on [0,infinity) we have that the integral of f(x) from 0 to 1 is finite and hence, adding the finite integral of f(x) from 1 to infinity we obtain that the integral of f(x) from 0 to infinity is also finite.