[proofplan] We show the equivalence of two formulations of [differentiability](/page/Derivative): the error-function formulation ($f(a + h) = f(a) + \tau(h) + |h|\varepsilon(h)$ with $\varepsilon$ continuous at $\mathbf{0}$ and $\varepsilon(\mathbf{0}) = \mathbf{0}$) and the limit formulation. The two are connected by defining $\varepsilon(h) = [f(a + h) - f(a) - \tau(h)]/|h|$ for $h \neq \mathbf{0}$ and $\varepsilon(\mathbf{0}) = \mathbf{0}$; the limit condition is exactly the statement that $\varepsilon$ is continuous at $\mathbf{0}$. [/proofplan] [step:Define the error function linking the two formulations] Define $\varepsilon: (U - a) \to \mathbb{R}^n$ by \begin{align*} \varepsilon(h) = \begin{cases} \dfrac{f(a + h) - f(a) - \tau(h)}{|h|} & \text{if } h \neq \mathbf{0}, \\begin{align*}6pt] \mathbf{0} & \text{if } h = \mathbf{0}. \end{cases} \end{align*} Here $U - a = \{h \in \mathbb{R}^m : a + h \in U\}$, which is an open subset of $\mathbb{R}^m$ containing $\mathbf{0}$ (since $a \in U$ and $U$ is open). For $h \neq \mathbf{0}$, rearranging gives \begin{align*} f(a + h) = f(a) + \tau(h) + |h|\varepsilon(h). \end{align*} This identity also holds at $h = \mathbf{0}$ (both sides equal $f(a)$). [/step] [step:Prove the forward direction: error-function differentiability implies the limit condition] Suppose $f$ is [differentiable](/page/Derivative) at $a$ with derivative $\tau$: there exists $\varepsilon$ continuous at $\mathbf{0}$ with $\varepsilon(\mathbf{0}) = \mathbf{0}$ and $f(a + h) = f(a) + \tau(h) + |h|\varepsilon(h)$. For $h \neq \mathbf{0}$, dividing by $|h|$: \begin{align*} \frac{f(a + h) - f(a) - \tau(h)}{|h|} = \varepsilon(h) \to \mathbf{0} \quad \text{as } h \to \mathbf{0}, \end{align*} since $\varepsilon$ is continuous at $\mathbf{0}$ with $\varepsilon(\mathbf{0}) = \mathbf{0}$. [/step] [step:Prove the reverse direction: the limit condition implies error-function differentiability] Suppose \begin{align*} \lim_{h \to \mathbf{0}} \frac{f(a + h) - f(a) - \tau(h)}{|h|} = \mathbf{0}. \end{align*} Define $\varepsilon$ as above: $\varepsilon(h) = [f(a + h) - f(a) - \tau(h)]/|h|$ for $h \neq \mathbf{0}$ and $\varepsilon(\mathbf{0}) = \mathbf{0}$. The limit condition states exactly that $\varepsilon(h) \to \mathbf{0}$ as $h \to \mathbf{0}$, so $\varepsilon$ is continuous at $\mathbf{0}$ with $\varepsilon(\mathbf{0}) = \mathbf{0}$. Rearranging: \begin{align*} f(a + h) = f(a) + \tau(h) + |h|\varepsilon(h), \end{align*} which is the [differentiability](/page/Derivative) condition with linear approximation $\tau$ and error function $\varepsilon$. [/step]