Show that the function f defined by
`f(x)={(1 if "x is rational"),(0if "x is irrational"):}`
is discontinuous `AAne0inalpha`

Explanation

Consider any real point x=a
If a is rational then f(a)=1.
Again `lim_(xtoa+)f(x)=lim_(hto0)f(a+h)`
which does not exist because a+h may be rational or irrational.
Similarly `lim_(xtoa-)f(x)` does not exist.
Thus f(x) is discontinuous at any rational point.Similarly we can show that f(x) is discontinuous at any rational point . Similarly we can show that f(x) is discontinuous at any irrational point .
Hence f(x) is discontinuous for all `xinR`