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Here are a couple of the functions which originally forced mathematicians to refine their ideas of continuity.

- Let
*f*be the function defined by*f*(*x*) = 1 if*x*is rational and*f*(*x*) = 0 if*x*is irrational.

Then*f*is discontinuous at*every*point*x*.**Proof**

Take*p*∈**Q**and let (*x*_{n}) be a sequence of*irrationals*converging to*p*. Then*f*(*p*) = 1 but*f*(*x*_{n}))→ 0 and so*f*is discontinuous at*p*.

Similarly, if*p*∉**Q**then choose a sequence of*rationals*converging to*p*and deduce the same result.

An even cleverer (and more horrible) function is the following amazing example due to Dirichlet (1805 to 1859). - Let
*f*be the function defined by*f*(*x*) = 0 if*x*is irrational and*f*(*x*) =^{1}/_{b}if*x*is the rational number^{a}/_{b}(in lowest terms).

Then*f*is discontinuous at every rational point, but continuous at every irrational point.**Proof**

Take*p*=^{a}/_{b}∈**Q**and let (*x*_{n}) be a sequence of*irrationals*converging to*p*. Then*f*(*p*) =^{1}/_{b}≠ the limit of*f*(*x*_{n})).

However, if*p*∉**Q**then given*ε*> 0, we may find an interval around*p*which misses all the rationals of the form^{a}/_{b}with^{1}/_{b}<*ε*. Then for sequences lying in this interval we*do*have (*f*(*x*_{n}))→ 0 =*f*(*p*) and so*f*is continuous at these points.

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