Define Fx integralx0 ftdt for x ge 0 where f 0infinity righ

Define F(x) = integral^x_0 f(t)dt for x ge 0, where f: [0,infinity) rightarrow R is continuous and f(t) ge 1 for all t ge 0. Prove that if c > 0, then there is a unique solution to F(x) = c, x > 0.

Solution

it is easy to see F(x) is an increasing function since F\'(x) = f(x) > 0

Now

F(0) = 0

and for large x

F(x) goes to infinity since F(x) >= x

Hence there must exist some x_0 such that F(x_0) = c (because F is conitnuous Roll\'s theorem)

this point is unique because F is strictly increasing

F(x) = F(y) => x = y