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In the proof, mean value theorem is used (in the equal signs following A). Hence, the conditions for the theorem to be true would be as follows:

1. ##\varphi(y)## is continuous in the domain ##[b, b+h]## and differentiable in the domain ##(b, b+h),## and hence ##f(x,y)## is continuous in the domain ##\{a, a+k\} \times [b, b+h]## and differentiable in the domain ##\{a, a+k\} \times (b, b+h)##.

2. ##f_y(x,y)## is continuous in the domain ##[a, a+k] \times \{b+h_1\}## and differentiable in the domain ##(a, a+k) \times \{b+h_1\}##.

3. ##f_x(x,y)## is continuous in the domain ##\{a+k_1\} \times [b, b+h]## and differentiable in the domain ##\{a+k_1\} \times (b, b+h)##.

4. Since there is a limit ##(h, k) \rightarrow (0, 0)##, I presume ##f(x,y), f_y(x,y)## and ##f_x(x,y)## have to be continuous in the rectangular domain ##[a, a+k] \times [b, b+h]## and differentiable in the rectangular domain ##(a, a+k) \times (b, b+h)##.

The condition stated in the theorem is that the mixed partial derivatives be continuous. How does that satisfy the 4 conditions mentioned above? Does continuity at a point imply continuity in the ##\epsilon(x, y)##-neighbourhood of the point? And does differentiability at a point imply differentiability in the ##\epsilon(x, y)##-neighbourhood of the point?

1. ##\varphi(y)## is continuous in the domain ##[b, b+h]## and differentiable in the domain ##(b, b+h),## and hence ##f(x,y)## is continuous in the domain ##\{a, a+k\} \times [b, b+h]## and differentiable in the domain ##\{a, a+k\} \times (b, b+h)##.

2. ##f_y(x,y)## is continuous in the domain ##[a, a+k] \times \{b+h_1\}## and differentiable in the domain ##(a, a+k) \times \{b+h_1\}##.

3. ##f_x(x,y)## is continuous in the domain ##\{a+k_1\} \times [b, b+h]## and differentiable in the domain ##\{a+k_1\} \times (b, b+h)##.

4. Since there is a limit ##(h, k) \rightarrow (0, 0)##, I presume ##f(x,y), f_y(x,y)## and ##f_x(x,y)## have to be continuous in the rectangular domain ##[a, a+k] \times [b, b+h]## and differentiable in the rectangular domain ##(a, a+k) \times (b, b+h)##.

The condition stated in the theorem is that the mixed partial derivatives be continuous. How does that satisfy the 4 conditions mentioned above? Does continuity at a point imply continuity in the ##\epsilon(x, y)##-neighbourhood of the point? And does differentiability at a point imply differentiability in the ##\epsilon(x, y)##-neighbourhood of the point?

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