The function f(x) = cot x is discontinuous on the set

Concept:

Let f(x) = \(\rm \frac {p(x)}{q(x)}\)

There are three conditions that need to be met by a function f(x) in order to be continuous at a number a. These are:

• f(a) is defined [you can’t have a hole in the function]
• \(\rm \lim_{x \to a} f(x)\) exists
• \(\rm \lim_{x \to a} f(x) = f(a)\)

Note:

if any of the three conditions of continuity is violated, the function is said to be discontinuous.

If sin x = 0 then x = nπ, n ∈ Z 

Calculation:

Given:f(x) = cot x

\(\rm \cot x = \frac{\cos x}{\sin x}\)

Check where denominator becomes zero

sin x = 0

x = nπ, n ∈ Z

∴ Given function is discontinuous at x = nπ

Hence, option (1) is correct.

Important Points

• When dealing with a rational expression in which both the numerator and denominator are continuous.
• The only points in which the rational expression will be discontinuous where denominator becomes zero.