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

*f(x)* is differentiable at *a*.
- There exists a function continuous at
*x*_{0} *φ(x)* such that *f'(x*_{0}) = φ(x_{0}) and *f(x) = f(x*_{0}) + φ(x)(x − x_{0}) as *x → x*_{0}.
- There exists a scalar
*λ* such that *f(x) = f(x*_{0}) + λ(x − x_{0}) + o(x − x_{0}) as *x → x*_{0}.

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