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

[Tompo]

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]

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.

 

[Ackbach]

You can also check out the Differential Calculus Tutorial, in post # 2.