# Real Analysis Prove discontinuous

• May 10th 2013, 06:53 PM
leonhart
Real Analysis Prove discontinuous
f: [0,1]--> R (reals)

f(x) = {1 if x= 1/n where nEN (element Naturals) , 0 otherwise}

Prove f is discontinuous at 0

I have not been able to make any progress on this problem and need help please.
• May 10th 2013, 10:06 PM
Gusbob
Re: Real Analysis Prove discontinuous
Quote:

Originally Posted by leonhart
f: [0,1]--> R (reals)

f(x) = {1 if x= 1/n where nEN (element Naturals) , 0 otherwise}

Prove f is discontinuous at 0

I have not been able to make any progress on this problem and need help please.

$\displaystyle \displaystyle{\lim_{n\to \infty}}f\left(\frac{1}{n}\right)= 1 \to 1 \not= f(0)=0$
• May 11th 2013, 03:16 AM
Plato
Re: Real Analysis Prove discontinuous
Quote:

Originally Posted by leonhart
f: [0,1]--> R (reals)
f(x) = {1 if x= 1/n where nEN (element Naturals) , 0 otherwise}
Prove f is discontinuous at 0

A word about notation. This problem assumes that $\displaystyle \mathbb{N}=\mathbb{Z}^+$, that is not always the case.

The function is $\displaystyle f(x)=\begin{cases}\dfrac{1}{x} &:\; x\in\mathbb{Z}^+\\0 &:\; \text{otherwise}\end{cases}$.

Note that $\displaystyle {\lim _{x \to {0^ + }}}f(x) = \infty$. Therefore $\displaystyle f$ cannot be continuous at $\displaystyle x=0$.

If you need a $\displaystyle \varepsilon;\delta$ proof note that $\displaystyle \left( {\forall \varepsilon > 0} \right)\left( {\exists J \in {\mathbb{Z}^ + }} \right)\left[ {0 < {J^{ - 1}} < \varepsilon } \right]$