Can we construct a function f:R→R such that it has intermediate value property and discontinuous everywhere?
I think it is probable because we can consider
y={sin(1x),if x≠0,0,if x=0.
This function has intermediate value property but is discontinuous on x=0.
Inspired by this example, let rn denote the rational number,and define
y={∑∞n=112n|sin(1x−rn)|,if x∉Q,0,if x∈Q.
It is easy to see this function is discontinuons if x is not a rational number. But I can't verify its intermediate value property.
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