# Epsilon-Delta proof for continuity of x^3 at x=1

#### Tompo

##### New member
I am trying to complete a previous exam and have come across a question which I am unable to do. I know how to complete an epsilon delta proof for limits, however, not to prove continuity.... We haven't seemed to cover this in our lecture notes :/

Using an epsilon-delta technique, prove that f(x) = x3
is continuous at x = 1

Can someone provide a brief proof of this explaining the steps?

#### Sudharaka

##### Well-known member
MHB Math Helper
I am trying to complete a previous exam and have come across a question which I am unable to do. I know how to complete an epsilon delta proof for limits, however, not to prove continuity.... We haven't seemed to cover this in our lecture notes :/

Using an epsilon-delta technique, prove that f(x) = x3
is continuous at x = 1

Can someone provide a brief proof of this explaining the steps?
Hi Tompo,

Welcome to MHB! First you have to be familiar with the epsilon delta definition of continuity (Refer >>this<<).

We say that the function $$f:I\rightarrow \Re$$ is continuous at $$c\in I$$ if for each $$\epsilon>0$$ there exists $$\delta>0$$ such that,

$| f(x) - f(c) |<\epsilon\mbox{ whenever }| x - c |<\delta$

In your case you have $$f(x)=x^3$$ and $$c=1$$. First take any $$\epsilon>0$$ and consider $$|f(x)-f(1)|$$. Try to find a $$\delta>0$$ such that $$|f(x)-f(1)|<\epsilon$$ whenever $$|x-1|<\delta$$. >>Here<< you will find some examples of using the epsilon delta definition to show continuity.

Kind Regards,
Sudharaka.

Staff member