# Ngô Quốc Anh

## June 11, 2021

### Second-order differentiability in terms of partial derivatives

Filed under: Uncategorized — Ngô Quốc Anh @ 0:28

Let $U \subset \mathbf R^n$ be open, $a \in U$ is an arbitrary point, and $f : U \to \mathbf R$ is a function. Recall that $f$ is called differentiable at $a$ if there is a linear map, denoted by $A$, such that

$\displaystyle \lim_{\|h\| \searrow 0} \frac{ \| f(a+h)-f(a) - A (h) \| }{\|h\|} = 0.$

The linear map $A$ is called derivative of $f$ at $a$, denoted by $f'(a)$. Notice that in the nominator of the above quotient, the symbol $\| \cdot \|$ is simply the absolute value function. But this is no longer true for higher-order derivatives that we are going to define.

The following theorem is well-known.

Theorem 1 (1st order differentiability). The function $f : U \to \mathbf R$ is differentiable at $a \in U$ if all partial derivatives $\partial_i f$ exist in a neighborhood of $a$ and are continuous at $a$.

When $f'(a)$ exists, we must have

$\displaystyle f'(a) (h) = \sum_{i=1}^n \partial_i f(a) h_i.$

In this note, we want to extend the above theorem for higher-order derivatives. To be more precise, we prove the following

Theorem 2 (2nd order differentiability). The function $f : U \to \mathbf R$ is twice differentiable at $a \in U$ if all partial derivatives $\partial_{ij} f$ exist in a neighborhood of $a$ and are continuous at $a$.

(more…)