Even when functions have a defined value at an input value, the function may be discontinuous. Limit of the function at that input value can be used to determine continuity.

*click on the content to continue..*

So far, the motivation to examine limits of a function was * to evaluate the function at a input value of the argument variable where the function evaluates to 'indeterminate value'*.

In this topic, another motivation to examine limits is explained.

Consider `f(x)=1/color(coral)(x-1)` at `x=1`

by directly substituting `x=1`

`f(1) = 1/(1-1) = +oo`

Consider `f(x)=1/color(coral)(x-1)`.

`lim_(x->1+)f(x)`

`quad quad = 1/color(coral)(1+delta-1) `

`quad quad = 1/color(coral)(delta)`

`quad quad = 1/0 = +oo`

Consider `f(x)=1/color(coral)(x-1)`

`lim_(x->1-)f(x)`

`quad quad = 1/color(coral)(1-delta-1) `

`quad quad = 1/color(coral)(-delta)`

`quad quad = -1/0 = -oo`.

For the function `f(x)=1/color(coral)(x-1)`,

• `f(a) = oo`

• `lim_(x->1+)f(x) = oo`

• `lim_(x->1-)f(x) = -oo`

The plot of the function is given in the figure.

That is, for a value less than `x=1`, the function is `-oo`. And at `x=1`, the function becomes `oo`. *The function is not continuous.*

A function `f(x)` at a given input value `x=a` is continuous if all the three are equal

`f(x)|_(x=a)`

`quad quad = lim_(x->a-) f(x)`

`quad quad = lim_(x->a+) f(x)`

The word 'continuous' means: unbroken and continue from one side to another without pause in between.

*familiarize with the terminology *

continuous

A function is *continuous* at an input value, if the function evaluated at the input equals both left-hand limit and right-hand limit of the function at that input value.

**Continuity of a Function: **A function `f(x)` is continuous at `x=a` if all the following three have a *defined value* and are *equal*

• Evaluated at the input value `f(x)|_(x=a)`

• left-hand-limit `lim_(x->a-) f(x)`

• right-hand-limit `lim_(x-a+) f(x)`

*Solved Exercise Problem: *

Given function `f(x) = 2x^2`, is it continuous at x=0?

- Yes, Continuous.
- Yes, Continuous.
- No, Not continous

The answer is 'Yes, Continuous'. Evaluate the three values of the function and they are equal.

*switch to interactive version*