f ( x ) = 0 if x is irrational, f ( x ) = 1 if x is rational. Does a limit exist?

Question

f  (  x  )  =  0 if   x is irrational,   f  (  x  )  =  1 if   x is rational. Does a limit exist?

Emma Singleton · Accepted Answer

Intuitively, a function being continuous means that if you nudge the input just a little bit, the output also changes just a little bit.
Let's consider

a = 5

. We know

f (5) = 1

. Now if you nudge the input by a very small irrational number

c

, we get

f (5 + c) = 0

, so the output,

0

, is relatively far from the output of our

a

value. The nail in the coffin is that you can take

c

to be as small as you want, and you will always get a relatively far output. This is exactly a property which discontinuous functions have.
With continuous functions on the other hand, if you make that

c

smaller and smaller, the distance between the outputs should get smaller and smaller too, not stay the same (or get bigger).

f(x)=0 if x is irrational, f(x)=1 if x is rational. Does a limit exist?

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