of_dslope — Mathlib

English

A simplication of the previous equivalence: hasDerivAt equivalence reduces to differentiability at a.

Русский

Упрощение предыдущего эквивалентности: эквивалентность hasDerivAt эквивалентна дифференцируемости в точке a.

LaTeX

$$$\text{ContinuousAt}( dslope\ f\ a, a) \iff \text{DifferentiableAt}\ f\ a$$$

Lean4

theorem of_dslope (h : ContinuousWithinAt (dslope f a) s b) : ContinuousWithinAt f s b :=
  by
  have : ContinuousWithinAt (fun x => (x - a) • dslope f a x + f a) s b :=
    ((continuousWithinAt_id.sub continuousWithinAt_const).smul h).add continuousWithinAt_const
  simpa only [sub_smul_dslope, sub_add_cancel] using this

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