If the function f is defined by

Question

f(x)=0 if x is rational
f(x)=1 if x is irrational

prove that lim x-->0 f(x) does not exist.

Can someone help me answer this question step by step so I would know how to prove a question like this next time? Thank you.

Answer 1

for the limit L to exist, |f(x)-L| < ε whenever |x| < δ

But no matter how small δ is, there are points in the interval [0,δ] where f=1 and f=0.