# One-Sided Continuity. Classification of Discontinuities

Similarly to the one-sided limits, we can define one-sided continuity.

Definition. Function f(x) is continuous from the right at point a if lim_(x->a^+)=f(a). Function f(x) is continuous from the left at point a if lim_(x->a^-)f(x)=f(a).

Clearly, if function is continuous from the left and from the right at point a, then it is continuous at point a.

Definition. Function f is discontinuous at a if it is not continuous.

There are three kinds of discontinuity at a:

1. Removable Discontinuity: if lim_(x->a)f(x) exists and finite, but function is either undefined at point a or lim_(x->a)f(x)!=f(a) . It is called removable because this discontinuity can be removed by redefining function as g(x)={(f(x) if x!=a),(lim_(x->a)f(x) if x=a):}
2. Jump or Step Discontinuity: if one-sided limits exist and finite but not equal.
3. Infinite or Essential Discontinuity: if one or both of the one-sided limits don't exist or are infinite.

Let's go through a couple of examples of these discontinuities.

Example 1. Find where function f(x)=(x^2+2x-3)/(x-1) is discontinuous and classify this discontinuity.

This function is rational, so it is continuous everywhere, except where denominator equals 0, i.e. where x-1=0. So, functions is not defined (and not continuous) when x=1.

Now, lim_(x->1)(x^2+2x-3)/(x-1)=lim_(x->1)((x+3)(x-1))/(x-1)=

=lim_(x->1)(x+3)=4.

Thus, limit exists and finite, but f(1) is not defined, so x=1 is removable discontinuity.

Example 2. Find where function f(x)={((x^2+2x-3)/(x-1) if x!=1),(2 if x=1):} is discontinuous and classify this discontinuity.

This is actually same example as example 1, except that function is defined at x=1.

Again lim_(x->1)(x^2+2x-3)/(x-1)=4, but f(1)=2, so lim_(x->1)(x^2+2x-3)/(x-1)!=f(1).

Thus, x=1 is point of removable discontinuity.

As in example 1 we can make function continuous by redefining function as f(x)={((x^2+2x-3)/(x-1) if x!=1),(4 if x=1):}.

Example 3. Find where function f(x)={(x+1 if x<=2),(4-x if x>2):} is discontinuous and classify this discontinuity.

Clearly linear functions are continuous everywhere, but since lim_(x->2^-)f(x)=lim_(x->2^-)(x+1)=3 and lim_(x->2^+)f(x)=lim_(x->2^+)(4-x)=2 then lim_(x->2^-)f(x)!=lim_(x->2^+)f(x).

This means that at x=2 there is jump discontinuity.

Also note, that lim_(x->2^-)f(x)=3=f(1), this means that function is continuous from the left at x=2.

Example 4. Find where function f(x)=1/(x-1) is discontinuous and classify this discontinuity.

This function is rational, so it is continuous everywhere, except where denominator equals 0, i.e. where x-1=0. So, functions is not continuous when x=1.

Now, lim_(x->1^+)1/(x-1)=oo.

This means that at x=1 there is infinite discontinuity.

Actually infinite discontinuity occurs when we have vertical asymptote.

Example 5. Find points where function f is discontinuous and classify these points: f(x)={(1/x if x<2),(1/x if 2<x<=4),(x if 4<x<6),(4 if x=6),(x if x>6):}

Function is discontinuous at 0 because lim_(x->0^+)f(x)=lim_(x->0^+)1/x=oo. So, at function x=0 we have infinite discontinuity.

Function is dicontinuous at 2 because f(2) is simply not defined. Since lim_(x->2^-)1/x=lim_(x->2^+)1/x=1/2 then lim_(x->2)1/x=1/2 and so x=2 is removable discontinuity.

Since lim_(x->4^-)f(x)=lim_(x->4^-)1/x=1/4 and lim_(x->4^+)f(x)=lim_(x->4^+)x=4 then lim_(x->4)f(x) doesn't exist because one-sided limits are not equal.

This means that x=4 is jump discontinuity.

Since lim_(x->6^-)f(x)=lim_(x->6^-)x=6 and lim_(x->6^+)f(x)=lim_(x->6^+)x=6 then lim_(x->6)f(x)=6 but f(6)=4!=6 , so lim_(x->6)f(x)!=f(6).

This means that x=6 is removable discontinuity.

Example 6. Conside function f(x)=[x] where [x] is floor function.

This function is discontinuous at every integer point x=n, because lim_(x->n^-)[x]=n-1 and lim_(x->n^+)[x]=n.

So one-sided limits are finite but not equal. This means that at every integer point x=n there is jump discontinuity.

Also, note that f(n)=n=lim_(x->n^+)f(x) that's why at every integer point x=n function is continuous from the right.