As Q is (infinitely) countable, we can find a bijection n→rn from N to Q. We now reuse the function f defined here. www.mathcounterexamples.net/a-differentiable-function-except-at-point-with-bounded-derivative Recall f main properties: 略 This proves that hh is differentiable at aa with h′(a)=limn→+∞h′n(a). For a∈Q, we can find p∈N with a=rp. Following a similar proof than above, the function lp:x→h(x)−up(x) is differentiable at a. As f does not have left and right derivatives at 00, upup does not have left and right derivatives at a. finally, the equality h=lp+up implies that hh also does not have left and right derivatives at a.
Conclusion: the function h is differentiable at all irrational points but does not have left or right derivative at all rational points. (引用終り)
