Let f:R^2->R be given by f(x,y) = x if y = x^2 and 0 otherwise. Then f is continuous at (0,0) and all directional derivatives there are zero but f is not differentiable at (0,0). In this example, we have non-differentiability even though we have a vector (the zero vector) that "behaves" like a gradient with respect to giving the directional derivatives upon taking inner products with the direction.
davidrosenberg
commented
7 years ago
Let f:R^2->R be given by f(x,y) = x if y = x^2 and 0 otherwise. Then f is continuous at (0,0) and all directional derivatives there are zero but f is not differentiable at (0,0). In this example, we have non-differentiability even though we have a vector (the zero vector) that "behaves" like a gradient with respect to giving the directional derivatives upon taking inner products with the direction.