Monday, 19 December 2016

real analysis - Can we construct a function f:mathbbRrightarrowmathbbR such that it has intermediate value property and discontinuous everywhere?


Can we construct a function f:RR such that it has intermediate value property and discontinuous everywhere?





I think it is probable because we can consider
y={sin(1x),if x0,0,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(1xrn)|,if xQ,0,if xQ.



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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