P3367
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. $\square$
