# Continuity of Special Types of Functions

**Continuity of functions in which**

*greatest integer*function is involved*f*(

*x*) = [

*x*] is discontinuous when

*x*is an integer.

Similarly

*f*(*x*) = [*g*(*x*)] is discontinuous when*g*(*x*) is an integer, but this is true only when*g*(*x*) is monotonic (*g*(*x*) is strictly increasing or strictly decreasing).For example,

*f*(*x*) = is discontinuous when is an integer, as is strictly increasing (monotonic function).*f*(

*x*) = [

*x*

^{2}],

*x*â‰¥ 0 is discontinuous when

*x*

^{2}is an integer, as

*x*

^{2}is strictly increasing for the

*x*â‰¥ 0.

Now consider

*f*(*x*) = [sin*x*],*x*âˆˆ [0, 2*Ï€*] function.*g*(*x*) = sin*x*is not monotonic in [0, 2*Ï€*]. For this type of function points of discontinuity can be determined easily by graphical methods. We can note that at*x*= 3*Ï€*/2, sin*x*takes integral value â€“1, but at*x*= 3*Ï€*/2,*f*(*x*) = [sin*x*] is continuous.**Continuity of functions which**We know that

*signum*function is involved*f*(

*x*) = sgn(

*x*) is discontinuous at

*x*= 0.

In general,

*f*(*x*) = sgn(*g*(*x*)) is discontinuous at*x*=*a*if*g*(*a*) = 0.**Continuity of functions involving**

^{}We know that

**Illustration**

Discuss continuity of

*f*(*x*) = .Solution

*f*(

*x*) = =

= =

Thus

*f*(*x*) is discontinuous at*x*= Â±1.**Continuity of functions in which**

*f*(*x*) is defined differently for rational and irrational values of*x***Illustration**

Discuss the continuity and discontinuity of the following function:

*f*(

*x*) =

Solution

For any

*x*=*a*,LHL = = 0 or 1
(as can be rational or irrational)

Similarly, RHL = = 0 or 1

Hence

*f*(*x*) oscillates between 0 and 1 as for all values of*a*.Therefore, LHL and RHL do not exist.

Hence

*f*(*x*) is discontinuous at a point*x*=*a*for all values of*a*.**Illustration**

Find the value of

*x*where*f*(*x*) = is continuous.Solution

*f*(

*x*) is continuous at some

*x*=

*a*, where

*x*= 1 â€“

*x*or

*x*= 1/2.

Hence

We have

*f*(*x*) is continuous at*x*= 1/2.**Explanation:**

*f*(1/2) = 1/2.

If

*x*â†’ 1/2^{+}then*x*may be rational or irrationalâ‡’

*f*(1/2^{+}) = 1/2 or 1 â€“ 1/2 = 1/2.If

*x*â†’ 1/2^{â€“}then*x*may be rational or irrationalâ‡’

*f*(1/2^{â€“}) = 1/2 or 1 â€“ 1/2 = 1/2.Hence

*f*(*x*) is continuous at*x*= 1/2. For some other point say*x*= 1.*f*(1) = 1

If

*x*â†’ 1^{+}then*x*may be rational or irrationalâ‡’

*f*(1^{+}) = 1 or 1 â€“ 1 = 0.Hence

*f*(1^{+}) oscillates between 1 and 0, which causes discontinuity at*x*= 1.Similarly,

*f*(*x*) oscillates between 0 and 1 for all*x*âˆˆ*R*â€“ {1/2}.