I’m working on a calculus question and need guidance to help me study.
3. In this exercise, we will prove Darboux’s Theorem: Suppose f : [a; b] ! R is continuous
on [a; b] and dierentiable on (a; b). Let a < x1 < x2 < b and suppose there is a L 2 R
such that f0(x1) < L < f0(x2) (or the other way around). Then, there exists x 2 (x1; x2)
such that f0(x) = L.
In particular, whilst the derivative may not be continuous it does satisfy the Intermediate
Value Property.
(a) Consider the function g(x) := f(x) Lx. Show that g is dierentiable on (a; b) and
g0(x1) < 0 < g0(x2).
(b) Argue that g must have a minimum at some point x 2 (x1; x2).
(c) Use part (b) to complete the proof.
13. Assume f : [a; b] ! R is Riemann integrable on [a; b].
(a) Show that if one value of f(x) is changed at some point x 2 [a; b], then f is still
integrable and integrates to the same value as before.
(b) Show that the observation in (a) remains if we change only a nite number of values
of f.
(c) Find an example to show that altering f on an innite number of points may cause
it cause the resulting function to no longer be Riemann integrable.
0 comments