Prove that the function: f ( x ) = { <mtable columnalign="left left"

Despiniosnt 2022-05-22 Answered
Prove that the function:
f ( x ) = { x ,  if  x  is an irrational number  0  if  x  is a rational number 
is discontinuous at every irrational number using both the precise definition of a limit and the fact that every nonempty open interval of real numbers contains both irrational and rational numbers.
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Answers (1)

Krish Finley
Answered 2022-05-23 Author has 14 answers
Let a be irrational, so f ( a ) = a 0. Let ϵ = | a | and assume there is δ > 0 such that | f ( x ) f ( a ) | < ϵ for all x with | x a | < δ. By the existence of rationals x with | x a | < δ (for example x = 1 n n a for x = 1 n n a with n > 1 δ ) we arrive at a contradiction because for this x we have f ( x ) = 0 and hence | f ( x ) f ( a ) | ϵ.
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