Saturday, 12 October 2019

real analysis - Does there exist a bijective differentiable function f:mathbbR+rightarrowmathbbR+, whose derivative is not a continuous function?





Does there exist a bijective differentiable function f:R+R+, whose derivative is not a continuous function?




x2sin1x is a good example for non-continuous derivative function, that will not work here, I guess.


Answer



The function
f(x):=x2(2+sin1x)+8x(x0),f(0):=0,


is differentiable and strictly increasing for x1, and its derivative is not continuous at x=0. Translate the graph of f one unit and eight units , and you have your example.


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