Section 11.9: restrict Exercise 11.9.3 to the open interval#534
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The equivalence
DifferentiableWithinAt ℝ (fun x => integ f (Icc a x)) (Icc a b) x₀
↔ ContinuousWithinAt f (Icc a b) x₀
fails at the endpoints, so x₀ must lie in `Ioo a b`, not `Icc a b`.
At an endpoint the within-derivative is one-sided, and for monotone f it
always exists (it equals the one-sided limit), whereas f need not be
one-sided-continuous there. E.g. on [a,b] take f a = 0 and f = 1 on (a,b]:
then F(x) = integ f (Icc a x) = x - a has F'(a+) = 1, so F is differentiable
within at a, yet f is not (right-)continuous at a.
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The equivalence
DifferentiableWithinAt ℝ (fun x => integ f (Icc a x)) (Icc a b) x₀
↔ ContinuousWithinAt f (Icc a b) x₀
fails at the endpoints, so x₀ must lie in
Ioo a b, notIcc a b.At an endpoint the within-derivative is one-sided, and for monotone f it always exists (it equals the one-sided limit), whereas f need not be one-sided-continuous there. E.g. on [a,b] take f a = 0 and f = 1 on (a,b]: then F(x) = integ f (Icc a x) = x - a has F'(a+) = 1, so F is differentiable within at a, yet f is not (right-)continuous at a.