Page 232 - DMTH201_Basic Mathematics-1

Unit 8: Continuity

Notes

Definition: For a function f(x) defined on a set S, we say that f(x) is continuous on S if f(x) is
continuous for all a S.
Example: We have seen that polynomial functions are continuous on the entire set of real
numbers. The same result holds for the trigonometric functions sin(x) and cos(x).

The following two exercises discuss a type of functions hard to visualize. But still one can study
their continuity properties.

Example: Discuss the continuity of
f(x) =

Solution:
Let us show that for any number a, the limit does not exist. Indeed, assume otherwise
that
= L.

Then from the definition of the limit implies that for any > 0, there exists > 0, such that
|x – a|< |f(x) – L|< .

Set Then exists > 0, such that

|x – a|<

or equivalently
a – < x < a – <

Since any open interval contains a rational and an irrational numbers, then we should have

Combining the two inequalities we get

which leads to an obvious contradiction. Thus, the function is discontinuous at every point a.

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