Let $f : \mathbb{R}^N \to \mathbb{R}^M$ be a
D4363: Euclidean real function such that
| (i) |
\begin{equation}
\exists \, C \in \mathbb{R}^M :
\forall \, x \in \mathbb{R}^N :
f(x) = C
\end{equation}
|
Let $L : \mathbb{R}^N \to \mathbb{R}^M$ be a
D4364: Real function such that
| (i) |
\begin{equation}
\forall \, x \in \mathbb{R}^N :
L(x) = (0, 0, \ldots, 0)
\end{equation}
|
For brevity, denote $(0, 0, \ldots, 0) \in \mathbb{R}^M$ by $0$. Let $\varepsilon > 0$. If $x, x_0 \in \mathbb{R}^N$ such that $x \neq x_0$, then we have
\begin{equation}
\frac{|f(x) - f(x_0) - L(x - x_0)|}{|x - x_0|}
= \frac{|C - C - 0|}{|x - x_0|}
= 0
< \varepsilon
\end{equation}
Since $\varepsilon > 0$ was arbitrary, the claim follows. $\square$
