Page 17 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes 1.4 Review Questions

1. If the function f assumes its maximum at c, show that D+ f (c) 0 and D f (c) 0.
–
+
+
+
2. Give an example of functions such that D (f + g) D f + D g.
3. Find the four Dini’s derivatives of function f : [0, 1]  R
such that f (x) = 0, if x 0, if x Q and f (x) = 1, if x Q.
4. Evaluate the four Dini’s derivative at x = 0 of the function f (x) given below:

2 1 2 1
ax sin bx cos , x 0
x
x
f (x) = 2 1 2 1
px sin qx cos , x 0
x x
and f (0) = 0, given that a < b, p < q.
5. Every point of continuity of an integrable function f (t) is a Lebesgue point of f (t). Elucidate.

1.5 Further Readings

Books J. Yeh, Real Analysis: Theory of Measure and Integration

Bartle, Robert G. (1976). The Elements of Real Analysis (second edition ed.)

Online links www.solitaryroad.com/c756.html
www.public.iastate.edu/.../Royden_Real_Analysis_Solutions.pdf

10 LOVELY PROFESSIONAL UNIVERSITY