Differentiability criterion via remainder estimate

Let $U\subseteq\mathbb{R}^n$ be open , let $f:U\to\mathbb{R}^m$, and fix $a\in U$.

Proposition (remainder estimate form of differentiability): The following are equivalent:

• $f$ is differentiable at $a$.
• There exists a linear map $A:\mathbb{R}^n\to\mathbb{R}^m$ such that $\lim_{h\to 0}\frac{\|f(a+h)-f(a)-Ah\|}{\|h\|}=0.$
• Equivalently: there exists a linear map $A$ such that for every $\varepsilon>0$ there exists $\delta>0$ with $0<\|h\|<\delta \implies \|f(a+h)-f(a)-Ah\|\le \varepsilon\,\|h\|.$

In this formulation, $Ah$ is the best linear approximation to $f$ near $a$ and the remainder is “small compared to $\|h\|$.”

Proof sketch: The second statement is the definition of differentiability. The third is exactly the $\varepsilon$–$\delta$ rewriting of the limit in the second statement: a limit equals $0$ iff it can be made $<\varepsilon$ for all sufficiently small $h$.