Page 173 - DMTH401_REAL ANALYSIS

Unit 12: Properties of Continuous Functions

Before we end this unit, we state a theorem without proof regarding the continuity of the Notes
inverse function of a continuous function.
Theorem 6: Inverse Function Theorem

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Let f : I – J be a function which is both one-one and onto. If f is continuous on I, then f : J  I is
continuous on J. For example the function.
é -  ù
f : , [–1, 1] defined by
ê ú
ë 2 2 û
f(x) = sin x,

é -  ù
is both one-one and onto. Besides f is continuous on , . Therefore, by Theorem 6, the
ê ú
ë 2 2 û
function
é -  ù
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f : – 11  ê , ú defined by
ë 2 2 û
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f (x) = sin –x
is continuous on [–1, 1].

Self Assessment

1. Give an example of the following:

(i) A function which is nowhere continuous but its absolute value is everywhere
continuous.

(ii) A function which is continuous at one point only.
(iii) A linear function which is continuous and satisfies the equation f(x + y) = f(x) + f(y).
(iv) Two uniform continuous functions whose product is not uniformly continuous.

2. State whether the following are true or false:
(i) A polynomial function is continuous at every point of its domain.

(ii) A rational function is continuous at every point at which it is defined.
(iii) If a function is continuous, then it is always uniformly continuous.
x
(iv) The functions e and log x are inverse functions for x > 0 and both are continuous for
each x > 0.
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(v) The functions cos x and cos x are continuous for all real x.
(vi) Every continuous function is bounded.
(vii) A continuous function is always monotonic.

(viii) The function sin x is monotonic as well as continuous for s [0, 3]
(ix) The function cos x is continuous as well as monotonic for every x  R.
(x) The function |x|, x  R is continuous.

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