Continuity with directional derivatives

Question

I face a problem in my exam If a function with two variables is there any relation between continuity and directional derivatives, please explain with example it's really helpful thankyou

Answer 1

There is no relation in the following sense: you can have a continuous function but whose partial derivatives do not exist, and conversely you can find a function whose partial derivatives exist but the function is not continuous. Examples for both cases can be found in the following questions:

Does existence of partial derivatives implies continuity at a point $(x_0,y_0)$?

Does continuity imply partial derivatives exist?