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

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Solution :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`


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