Is there a function which is infinitely-differentiable everywhere but continuous nowhere?
EDIT: The graphic which has since been deleted was taken from my own notes, and I can clarify what definition of "differentiable" was being used there (presumably, the meaning the OP had in mind was the same).
A function is differentiable at $x$ iff (i) all directional derivatives at $x$ exist, and (ii) the map that sends a tangent vector to the corresponding directional derivative is continuous and linear. (Of course, in finite dimensions, the condition "continuous" is superfluous.) This is strictly stronger than gâteaux differentiable and strictly weaker than fréchet differentiable. (FWIW, that this definition is nonstandard is made clear immediately following the definition, where it is contrasted with both gâteaux differentiability and fréchet differentiability.)
If you're curious, the motivation for this terminology is as follows.
Using all three terms allows me to be more precise, and as "fréchet differentiable" and "gâteaux differentiable" already have names, by necessity the above condition is referred to as simply "differentiable" (which I found more palatable than just making up a new term).
This definition is easier (and, IMHO, more natural) than fréchet differentiable, and furthermore, almost everything that is true for fréchet differentiable functions is true of functions which are differentiable in this sense (the fact in question here being the biggest exception I am aware of).
While this is (slightly) more difficult than the definition of gâteaux differentiable, I found that not having the derivative be a one-form 'broke' things to an unacceptable degree (for example, as I only defined the derivative of tensor fields, if the derivative itself were not a tensor field, then strictly speaking the second derivative would have been left undefined).