Gateaux but not Frechet differentiable functional For functional between Banach spaces X,Y:
By Gateaux differentiable at $u\in X$ I mean that there exists bounded linear operator $dF(u)$ s.t. $F(u+t\xi)-F(u)=dF(u)\xi+o(t)$ for all $\xi\in X$.
For Frechet we have $F(u+\xi)-F(u)=dF(u)\xi+o(\left \|  \xi\right \|)$ uniformly for $\xi\in S_{X}$ (i.e. for $\xi$ with same length).
For example $f(x,y)=0$ for $(x,y)=0$ and $f(x,y)=\frac{x^{2}y}{x^{6}+y^{2}}$ is Gateaux differentiable but not Frechet. A more difficult example is $F:L^{1}([0,\pi])\to \mathbb{R}$ defined as $F(u)=\int_{0}^{\pi}sin(u(t)) dt$ [1].
Any other interesting examples, eg. functionals acting on Sobolev spaces or $L^{p}(\Omega)$? The shorter the proofs the better because I want to present them to class.
Thank you
[1]http://www.m-hikari.com/ams/ams-password-2009/ams-password17-20-2009/gaxiolaAMS17-20-2009.pdf

It may  require too much background for your class presentation, but you might be interested in an application of this to statistical mechanics.  See my paper
with R.R. Phelps: "Some convexity questions arising in statistical mechanics," Math. Scand. 54(1984), 133-156. The "pressure" is a real-valued convex function on a Banach space of "interactions"; under certain assumptions, it is Gateaux differentiable on a dense $G_\delta$ set, but nowhere Frechet differentiable. 
