Deduce form
Δf = (f_x(a,b))Δx + (f_y(a,b))Δy + (ε_1)Δx + (ε_2)Δy
(f_x(a,b) and f_y(a,b)are partial derivatives of f with respect to x and y)
that under the hypotheses of the linear approximation theorem, Δf approaches 0 as Δx approaches 0 and Δy approaches 0. What does this imply about the continuity of f at the P(a,b)?
Would the answer be that f is continuous at P(a,b) because as delta x, y approach 0, f approaches 0. And applying the epsilon delta definition says that a limit exist. Since the partial derivatives exist at P, then f must also. Thus f is continuous at P.