A Property of Maximum Function

Given functions $F = \{f_k\}_{k=1}^{n}$, for any fixed $t$ define the maximum function as $f(t) = \max_{f_k \in F} f_k(t)$, and set $\Pi = \{f_k \in F: f_k(t) = f(t) \}$, we have if $f^\prime$ exists at $t$, then $f^\prime(t) = f_{\pi \in \Pi}^\prime(t)$.

proof: [General idea: $A = B \Leftrightarrow A \leq B \& A \geq B$]

• $f^\prime(t) \geq f_{\pi \in \Pi}^\prime(t)$:

$$\begin{aligned} f^\prime(t) &= f^\prime(t+) \\ &= \lim_{x \rightarrow t+}\frac{f(x) - f(t)}{x - t}\\ \end{aligned}$$

Since $f(x) \geq f_{\pi}(x)$, $f(t) = f_{\pi}(t)$ for any $\pi \in \Pi$, and $x > t$,

$$\begin{aligned} f^\prime(t) &\geq \lim_{x \rightarrow t+}\frac{f_{\pi}(x) - f_{\pi}(t)}{x - t}\\ &= f_{\pi}^\prime(t+)\\ &= f_{\pi}^\prime(t) \end{aligned}$$

• $f^\prime(t) \leq f_{\pi \in \Pi}^\prime(t)$:

$$\begin{aligned} f^\prime(t) &= f^\prime(t-) \\ &= \lim_{x \rightarrow t-}\frac{f(x) - f(t)}{x - t}\\ &\leq \lim_{x \rightarrow t-}\frac{f_{\pi}(x) - f_{\pi}(t)}{x - t}\\ &= f_{\pi}^\prime(t-)\\ &= f_{\pi}^\prime(t) \end{aligned}$$