Problem
17: 1998-1999
Given 
(a) Find 
(b) Is f continuous on its domain? What does this mean about
f -1(x)?
Difficulty rating: 
Solution:
(a) This problem is not as hard as it looks! We first need to
find f-1(0). In other words, we want to know when f(x)
equals zero (see Problem 16 for further explanation of my reasoning). An accumulation
function will equal zero whenever the result of the integral is zero (in this
case when the upper and lower bounds for integration are equal, or when x
= 2). Thus, f -1(0)=2. Now, we need to find 1/f '(f
-1(0))--or the reciprocal of f '(2), which as we just
said is f -1(0). Note that
So, we evaluate the derivative at x = 2 and get
(b) Yes! The function is continuous
on its domain; its values are achived by accumulating area beneath the
crazy fraction, which does not have a point of discontinuity. If a function
is continuous on its entire domain, so is its inverse.
