I am aware of several examples showing that existence of directional derivative in all directions does not imply differentiability at a point, but why is the existence of directional derivatives not enough to assure differentiability?
For example, I understand the idea and relation between smoothness and differentiability, but how come smoothness cannot be guaranteed when we just know directional derivatives exist?
Hope I am clear about my concern.
Thanks!
The only good answer to this is because counterexamples exist. The prime examples being a non-linear homogeneous function defined away from the origin (e.g $f(x,y)=\frac{x^2y}{x^2+y^2}$ and $f(0,0)=0$).
I can flip your question on its head. Why do you expect one-dimensional information (existence of directional derivative) to give you information about all directions simultaneously (existence of derivative)? Even if you know all the directional derivatives exist, so what? Homogeneous functions are perfect examples of such bad behavior because along each direction, they behave like a linear function, which is as nice as things can get. But once you take all directions into account simultaneously (not one at a time), things may fall apart (as you see in the nonlinear homogeneous function example).
A more ‘poetic phrasing’: you can’t just stick any random bunch of talented artists together and expect them to produce a masterpiece.