Page 33 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Monotonic Decreasing Function: A monotonic decreasing function is a function that either
decreases or remains the same, never increases i.e. a function f(x) such that f(x ) f(x ) for x > x .
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Monotonic Function: A monotonic function is a function that is either a monotonic increasing or
monotonic decreasing.
Monotonic Increasing Function: A monotonic increasing function is a function that either
increases or remains the same, never decreases i.e. a function f(x) such that f(x ) f(x ) for x > x .
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2.4 Review Questions

1. Show that sum and product of two functions of bounded variation is again a function of
bounded variation.
2. Show that the function f defined on [0,1] by
x
xcos for 0 x 1
f x 2
0 for x 0
is continuous but not of bounded variation on [0,1].

3. Show that the function f defined on [0,1] as f(x) = xsin for x > 0, f(0)=0 is continuous but
x
is not of bounded variation on [0,1].
4. Define a function of bounded variation on [a,b]. Show that every increasing function on
[a,b] is of bounded variation and every function of bounded variation on [a,b] is
differentiable on [a,b].

5. Show that a continuous function may not be of bounded variation.
6. Show that a function of bounded variation may not be continuous.
7. If f is a function such that its derivative f’ exists and is bounded. Then prove that the
function f is of bounded variation.

2.5 Further Readings

Books Halmos, Paul (1950), Measure Theory, Van Nostrand and Co.

Kolmogorov, Andrej N.; Fomin, Sergej V. (1969). Introductory Real Analysis, New
York: Dovers Publications.

Online links www.ams.org

www.whitman.edu/mathematics/SeniorProjectArchive/.../grady.pdf

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