**We are now going to look at the two main types of discontinuities that can arise in a function. You should be able to distinguish between each type of discontinuity when a functionÂ fÂ may contain each type of discontinuity.**

**Discontinuity is of two kinds listed as,**

**(A) Discontinuity of 1st kind:**

(i) First kind removable discontinuity

(ii) Non-removable discontinuityÂ or jump discontinuity

**(i) First kind removable discontinuity**

IfÂ

then f(x) is said to have first kind of removable discontinuity:

**Example : Examine the function**

**Solution:**

Graphically,f(x) could be plotted as ,

Which showsÂ

Thus ,f(x) could be made continuously taking.

so, we could say f(x) becomes continuous, if

**(ii) Non-Removable discontinuity:**

Then f(x) is said to have First kind non-removable discontinuity.

**Example : Show the function,**

has non-removable discontinuity at x=0

**Solution:** We have

then R.H.L. at x=0, Let x=0+h

again, L.H.L. at x=0 , at x=0, Let x=0-h

\qquad \left[ as\quad h\longrightarrow 0;{ e }^{ -\frac { 1 }{ h } }\longrightarrow 0 \right] \\ \Rightarrow \lim _{ x\longrightarrow { 0 }^{ + } }{ f\left( x \right) } \neq \lim _{ x\longrightarrow { 0 }^{ – } }{ f\left( x \right) } &s=1 $

Thus,f(x) has non-removable discontinuity.

## (B)Discontinuity of 2nd kind:

If at least one ofÂ

is non- existent or infinite then f(x) is said to have discontinuity and 2nd kind at x=a

**Example : Show the function,**

Â has discontinuity at 2nd kind at x=0

**Solution:** We have

Which shows function has discontinuity of 2nd kind.

**Graphically:**

Here the graph is broken at x=0 as

Here the graph is broken at x=0 asÂ

Therefore f(x) is discontinuity of 2nd kind.