Transcribed Text
1. (Bounded Monotone). Let f(x) be a bounded monotone increasing function on an interval (a,b).
In this problem we prove that lim f(x) exists. (A similar argument could show lim f(x) also exists.)
b
(Note:
f
fxx is not necessarily defined or continuous at x = a, b. Recall that bounded means there exists an M
so that f(x) < M for all x. Monotone increasing means that x1 < x2
a) Let A = {f(x)  x € (a,6)}. Justify why this has a greatest lower bound Y.
b)
Explain
why
for
any
€ > 0 there must be an xo € (a,b) with f (xo) < Y + E. Use xo to find a 8 so
that 0  rl < € and complete the proof.
c) Explain in words why this means that for bounded monotone functions on (a,6), at each point
where c € (a,b) both lim f (x) and lim f(x) will exist. (Actually it can be shown that monotone
functions can only have a countable number of discontinuities. Since countable sets have measure 0
this means monotone functions will be continuous almost everywhere. Very nicely behaved!)
2. (Total Separation). Let f(x) be continuous on [a, b]. Suppose there is a value L such that f (x) # L
for any x € [a,b]. Complete the following steps to prove that there is an € > 0 such that f(x)  L > E
for all x € [a, b].
a) Justify why either L > M or L < m where M and m is the max and min of f(x) on [a,b].
b) Use L and M to construct an E and show that f (x)  E for all x € [a, b] (and similarly for
L
and m).
c) Show that the theorem fails if we replace [a, b] with the open interval (a,b).
3. (An Increasing derivative). Let f be differentiable on an interval I and f' (x) is increasing on I.
a) Let T(x) be the tangent line at a point c in the interval. Write down the equation for T(x).
b) Show for any x € I with x > c that f(x) > T(x).
(Hint: consider the auriliary function h(x) = f(x)  T(x). Apply the MVT f on [c, x] and use what you
know about f.)
c) Show that the same holds true for x € I with x < c. That is f(x) > T(x). Thus when f'(x) is
increasing the tangent line always lies below the function.
4. (Lipschitz again). Suppose f satisfies a Lipschitz condition of the form f(x)  f(y) < Kx  y on
the interval [a,b]. We prove that f is integrable on [a,b].
a) Prove that f must be bounded on [a,b].
(Hint: Use the condition to show that f(a)  K(b  a) < f(x) < f (a) + K(ba).) 
b) Use the Lipschitz condition on the each subinterval xi] to show that f (xi) + K Ari is an upper
bound for f on and f(xi)  Kxi is a lower bound for f on [xi1,xi]. Conclude that

€
c) Now given an € > 0 choose a partition Q of [a, b] so that each Axi Write
out
U ((Q)  L(Q) and, using part b, show that this is less than €.
d) What theorem allows you to conclude that f is integrable on [a.b)?
5. a) (Reciprocal Integrability). Let f be integrable on [a,b]. Give an example to illustrate that in
1
general
may not be integrable on [a,b].
f (x)
b)
Now assume f has a positive lower bound (i.e. f(x) > K > O for r € [a,b]). Using familiar notation
let
(i) M{ = sup {f(x) € [xi1,xi]}  and =
= x € xi and 1/f

Show that 0 < < 0 1 1 this implies
1
1
<
c) Use this inequality to show that  1 
1
d) Use the inequality in part to show that if f is integrable then is also integrable.
c
f(x)
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