1. The problem statement, all variables and given/known data The statement that the simple surface F has gradient at the ordered pair ((x,y),F(x,y)) of F means that there exists only one ordered number pair (p,q) such that if c is a positive number then there exists a rectangular segment , S, containing (x,y) such that, if (u,v) is in S and in the domain of F then F(u,v) - F(x,y) = (u-x)p + (v-y)q + ε|(u,v)-(x,y)| where ε is a number between -c and c 2. Relevant equations none 3. The attempt at a solution I realize that this is not a 'problem' per se, however I am simply trying to make sense of the definition of gradient without resorting to derivatives. I would like to prove that if F has gradient at a point then it is continuous at the point, again without using any notion of derivative, just the definition provided. But I would like to attempt to prove the implication of continuity myself. I am simply looking for anyone to assist me to break down the definition of gradient given and understand it a little better. I have been staring at it for a few days and I am just not getting anywhere. I don't understand where the positive number c comes into play and therefore the ε as well. Any assistance/insight/guidance would be appreciated, thanks!