Page 164 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 2. Examine the continuity of the function f: R ® R defined as,
3
(i) f(x) = x at a point a  R;
2
ì x - 4
ï , if x  2
(ii) f(x) = x - 2
í
ï 1, if x = 2
î
3. Show that the function f: R ® R defined by
ì 1, if x is rational
f(x) = í
î 0, if x is irrational
is totally discontinuous. Does f(a+) and f(a–) exist at any point a  R?
4. Prove that the function |f| defined by |f|(x) = |f(x)| for every real x is continuous on R
whenever f is continuous on R.
5. (i) Find the type of discontinuity at x = 0 of the function f defined by
f(x) = x + 1 if x > 0, f(x) = – (x + 1) if x < 0 and f(0) = 0.
(ii) The function f is defined by
1
f(x) = sin , x  0
x
= 0, x = 0
Is f continuous at 0?

Answers: Self Assessment

1. continuous 2. converges

3. real-valued function 4. h(x) = f(g(x))
5. continuous

11.7 Further Readings

Books Walter Rudin: Principles of Mathematical Analysis (3rd edition), Ch. 2, Ch. 3.
(3.1-3.12), Ch. 6 (6.1 - 6.22), Ch.7(7.1 - 7.27), Ch. 8 (8.1- 8.5, 8.17 - 8.22).
G.F. Simmons: Introduction to Topology and Modern Analysis, Ch. 2(9-13),
Appendix 1, p. 337-338.

Shanti Narayan: A Course of Mathematical Analysis, 4.81-4.86, 9.1-9.9, Ch.10,Ch.14,
Ch.15(15.2, 15.3, 15.4)
T.M. Apostol : Mathematical Analysis, (2nd Edition) 7.30 and 7.31.

S.C. Malik : Mathematical Analysis.
H.L. Royden : Real Analysis, Ch. 3, 4.

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