Directional derivatives in any direction do not imply continuity?

Question

I found an example where a function from R^2toR has directional derivatives at a point p at any direction however the function isn't continuous at p. I found this very weird because I thought that since our function has directional derivatives in any direction then is continuous in any direction and therefore in any direction I approach p the limit will be f(p). In other words it makes sense to me that a function f from R^2toR is continuous at a point p if and only if for any k the limit of the sequence f(p+1/n,p+k/n) as n tend to infinity is f(p).
So now I am wondering should the above be only if instead of if and only if? or did I understand wrong the definition of directional derivative? i.e my claim that since our function has directional derivatives in any direction then is continuous in any direction is false
Thank, You

Answer 1

Having directional derivatives in every direction does not imply continuity. It implies the function looks continuous from any direction, but you don't have to come at the point from one direction. You can come at it along loops, zigzags, or other crazy paths, too.

You don't seem to have misunderstood the definition of directional derivatives, but your understanding of what they tell you is off.