# Thread: Real Analysis Prove discontinuous

1. ## 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.

2. ## Re: Real Analysis Prove discontinuous

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.
$\displaystyle{\lim_{n\to \infty}}f\left(\frac{1}{n}\right)= 1 \to 1 \not= f(0)=0$

3. ## Re: Real Analysis Prove discontinuous

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 $\mathbb{N}=\mathbb{Z}^+$, that is not always the case.

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

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

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