Introduction to Analysis: Fall 2004
Practice problems VII
MTH 4101/5101 12/7/2004
1. Give an example of a non continuous function fthat is defined on [a, b] and
(i) satisfies the intermediate value property. (ii) does not satisy the
intermediate value property.
2. Suppose that f:[a, b]→Qis continuous on [a, b]. Prove that fis constant on
[a, b].
3. If a function f:[a, b]→IR is a continuous function which is one-one, then
prove that f−1:Rf→[a, b] is continuous.
4. Give an example of a function f:I→IR which is an injection, but its inverse
f−1is not continuous.
5. If a function fis a Lipschitz function, then show that it is uniformly
continuous.
6. Let Dbe a closed and bounded subset of IR and f:D→IR is a continuous
function. Prove that fis uniformly continuous on D.
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