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The Weierstrass-Carathéodory criterion for differentiation states that the following are equivalent:

  • f(x) is differentiable at a.
  • There exists a function continuous at x0 φ(x) such that f'(x0) = φ(x0) and f(x) = f(x0) + φ(x)(x − x0) as x → x0.
  • There exists a scalar λ such that f(x) = f(x0) + λ(x − x0) + o(x − x0) as x → x0.

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