Prove f(x) Continuous at x=1/2 for x Rational, Irrational

[ashok vardhan]

f[x]=x,x is rational, 1-x, x is irrational. prove that f(x) is only continuous at x=1/2.

 

[ashok vardhan]

please give a proper mathematical solution to this

 

[HallsofIvy]

Why? It looks like homework to me. You need to try yourself, first, and show us what you have.

Here's a nudge: for any x, there exist a sequence of rational numbers converging to x and there exist a sequence or irrational numbers converging to x.

 

[ashok vardhan]

i have already solved the problem in the way you suggested.but i have a problem in solving it using epsilon and those things

 

[Redbelly98]

Reminder: members are expected to show their attempt at solving a problem before they receive help.

ashok vardhan: please see your Private Messages for an important message, if you have not already done so.

 

[ashok vardhan]

f(x+y)=f(x)+f(y).f is continuous at x=0.prove that f(kx)=kf(x). i have proved it for k os an integer.for k a rational number i assumed it to be of p/q.and i can't proceed further to prove this. would you like to help in this

 

[HallsofIvy]

Is this a completely new problem? f(x+ y) is definitely NOT equal to f(x)+ f(y) for the problem you gave before. For example, $1+ \sqrt{2}$ is irrational and so $f(1+ \sqrt{2})= 1- (1+ \sqrt{2})= -\sqrt{2}$ but since 1 is rational and $\sqrt{2}$ is irrational, $f(1)+ f(\sqrt{2})= 1+ (1- \sqrt{2})= 2- \sqrt{2}$.

You didn't say anything about using $\epsilon$ and $\delta$ in your first post. If you are not allowed to use "$\lim_{x\to a} f(x)= L$ if and only if, for any sequence $\{x_n\}$ that converges to a, the sequence $\{f(x_n)\}$ converges to L", then copy the proof of that theorem, for this particular function.