Let f : R → R with intermediate value property and increasing over R \...

Wronsonia8g

Wronsonia8g

Answered

2022-06-30

Let f : R R with intermediate value property and increasing over R \ Q . Then f is continous on R . How to try?

Answer & Explanation

pampatsha

pampatsha

Expert

2022-07-01Added 15 answers

Let x be real. Take an irrational number α such that α > x. Clealry f ( α ) f ( x ), for if not, then
( f ( α ) , f ( x ) ) f [ ( x , α ) ] Q Q
by our hypotesis, which is absurd. By the same argument f ( α ) f ( x ) if α is irrational and α < x. Hence f is increasing on the whole line. Can you take it from here?

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