Linear Algebra
11.48 min read

Continuously Differentiable Functions of Many Variables

A function f:Rn→Rf: \mathbb{R}^n \to \mathbb{R} is called C1C^1 (continuously differentiable) if all nn partial derivative functions ∂f/∂x1,…,∂f/∂xn\partial f/\partial x_1, \ldots, \partial f/\partial x_n exist and are continuous everywhere on the domain.

This is a stronger condition than merely having all partial derivatives exist. A function can have all partial derivatives at a point without being differentiable or even continuous there — but if the partial derivatives are continuous near a point, then ff is automatically differentiable (and in fact has a well-defined local linear approximation) at that point.

The set of all C1C^1 functions on a domain forms a vector space. Compositions, sums, products, and quotients (away from zeros) of C1C^1 functions are C1C^1. This makes the C1C^1 class much easier to work with in practice than the bare "differentiable" class.

Formal View

Definition 11.6 — Class C¹ (Many Variables)
A function f:D⊆Rn→Rf: D \subseteq \mathbb{R}^n \to \mathbb{R} is of class $C^1$ on DD if all partial derivative functions ∂f∂x1,…,∂f∂xn\frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_n} exist and are continuous on DD. We write f∈C1(D)f \in C^1(D).
Theorem 11.1 — C¹ Implies Differentiability
If f∈C1f \in C^1 at a\mathbf{a}, then ff is differentiable at a\mathbf{a}, meaning a local linear approximation exists and the approximation error is o(∥h∥)o(\|\mathbf{h}\|).

The converse is false: a function can be differentiable without having continuous partial derivatives.

Why This Matters

The C1C^1 class is the standard regularity assumption in optimization theory and numerical analysis.

  • Guaranteeing gradient descent converges by ensuring gradients vary smoothly
  • Applying the inverse function theorem and implicit function theorem (both require C1C^1)
  • Ensuring that numerical differentiation (finite differences) converges to the true gradient

Quiz

Question 1

If all partial derivatives of ff exist at a point, then ff is C1C^1 at that point.

Question 2

Which condition is the strongest?

Common Mistakes

  • Confusing "all partials exist" with "C1C^1" — existence is much weaker than continuity of the partials.
  • Thinking C1C^1 and differentiability are the same — C1C^1 is strictly stronger.
  • Forgetting that the implication goes one way: C1C^1 gives differentiability, but differentiability does not give C1C^1.