A functional $\Phi$ is differentiable if there exist $F$ and $R$ such that $\Phi(f+h)-\Phi(f)=F(f,h)+R(f,h)$, where $F$ depends linearly on $h$ and $R(f,h) = O(h^2)$.
Define a functional $\Phi(f) = \max_{[0,1]} f$ where $f \in C([0,1])$, then is $\Phi$ differentiable?
I'm going to pretend $a=0$ and $b=1$ so we will be looking at $C([0,1]\to\mathbb{R})$.
Let's assume that there do exist such an $F$ and $R$. Then $F(0,h+k) = \Phi(h+k)-R(0,h+k)$ and $F(0,h+k) = \Phi(h) + \Phi(k) - R(0,h) - R(0,k)$, but $F(0,h+k) = F(0,h)+F(0,k)$, so you learn that $ R(0,h) + R(0,k) -R(0, h+k) = \Phi(h)+\Phi(k)-\Phi(h+k)$. Let $C > 0$ and let $h(x) = C(1-2x)$ and $k(x)=C(2x-1)$ so $h+k \equiv 0$. Noting that $0 = R(0,0)$ by assumption and $R(0,0)=R(0,h+k)=0$, you get $R(0,k) + R(0,h) = 2C$; in particular $R(0,k)+R(0,h) \neq o(C)$. However, by assumption, $R(0,k)=o(C)$ and $R(0,h)=o(C)$, which causes a problem since $o(C)+o(C)=o(C)$.