Question

(d)
Continuity at (a,b) is - condition for a
function f (x,y) to be differentiable at a point
(a,b) of its domain.​

Answer 1

Answer:

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Answer 2

Answer:

Let $D$ be the domain of $f(x,y)$ and $(a,b)\in D$. A funtion $f(x,y)$ is said to be diferentiable at $(a,b)$ if there exists a linear map $T(x,y)$ such that

$\lim_{(h,k) \to (0,0)} \frac{f(a+h,b+k)-f(a,b)-T(h,k)}{\|(h,k)\|} = 0$

It's very useful to use the following theorem to prove that functions are differentiable in a given point $(a,b)$: If the partial derivatives of $f$ are continuous in $(a,b)$, then $f$ is differentiable in $(a,b)$.