Contradiction of gradient between direction of steepest increase and local minimum

Question

I can understand the proof that at a local minimum, the gradient of that function must be zero. But I can't understand it together with the fact that gradient points the steepest increase, since if it is at local minimum, then moving at any direction will give the function some increment. So what does gradient being zero at this point means? If any direction increases this function, then why don't we just choose the direction with the greatest increment but choose zero which means not to move at all?

Answer 1

if the function is differentiable at a point of local minimum any partial derivative is equal to zero (as in the case of the function of one variable) thus the gradient is zero