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(x≠0),f(0):=0,
is differentiable and strictly increasing for x≥−1, 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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