Thursday, 13 June 2013

calculus - Is composition of surjective continuous function with discontinuous function discontinuous?

Let I1,I2,I3 be intervals R. Suppose f:I1I2 is a surjective continuous function and g:I2I3 is a discontinuous function. Must the composition gf be discontinuous?



There are some easy counter-examples if f is not assumed to be surjective, e.g. taking f to be a constant function, or in a way that "dodges" the discontinuous point(s) of g.



However if such "dodging" is prohibited, I fail to construct such functions nor find an answer from many similar questions on this site. So I am interested to know whether counter-examples exist? If not, is there a proof? Does it have something to do with intermediate value theorem?

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