Does there exist a function f:R→R that is strictly increasing and discontinuous everywhere?
My line of thought (possibly incorrect): I know there are increasing functions such as f(x)=x, and there are everywhere-discontinuous functions such as the Dirichlet function. I also know that when there is a discontinuity at a point c, there is a finite gap ϵ such that there are points d arbitrarily close to c such that |f(d)−f(c)|>ϵ. This is where my thinking gets unclear - does it make sense to have a "gap" at every real number?
Answer
There is no such function. Suppose that f:R→R is strictly increasing. For each a∈R let f−(a)= limx→a−f(x) and f+(a)=limx→a+f(x). Then f is discontinuous at a if and only if f−(a)<f+(a). Let D={a∈R:f is not continuous at a}, and for each a∈D let qa be a rational number in the non-empty open interval Ia=(f−(a),f+(a)).
It’s not hard to check that if a,b∈D with $a
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