NETMaths
Part CCSIR NET December 2023

Partial derivatives, differentiability, chain rule

For real numbers a, b, c, d, e, f, consider the function given by F(x, y) = (ax + by + c, dx + ey + f). Which of the following statements are true?

  1. A.F is continuous.
  2. B.F is uniformly continuous.
  3. C.F is differentiable.
  4. D.F has partial derivatives of all orders.

Solution

An affine map is Lipschitz (hence uniformly continuous), differentiable with constant derivative, and all higher partials are zero.

Related counterexample: If all partial derivatives exist at a point then f is continuous there

More on this topic

From Functions of Several VariablesPartial derivatives, differentiability, chain rule

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