Leta)f(x, y) is continuous at originb)f(x, y) is differentiable at originc)fx(0,0) ≠ 0d)fy(0 ,0) ≠ 0Correct answer is option 'A'. Can you explain this answer?

Question

Leta)f(x, y) is continuous at originb)f(x, y) is differentiable at originc)fx(0,0) &ne; 0d)fy(0 ,0) &ne; 0Correct answer is option 'A'. Can you explain this answer?

Answer

We are given that

Let us take ε > 0 and x² + y² ≠ 0.
Now consider,

We know that,

or equivalently

Thus,
| f(x ,y )-0 | ≤ | y | = 0- |x| + 1 • |y|
take

Therefore, for given ε > 0, there exists δ > 0 such that
|f(x, y) - 0 | < ε. whenever 0<|x|<δ,
0< |y |< δ
Hence,

Thus, f(x,y) is continuous at (0, 0).
Hence, option (a) is correct.
Now,
f(0 + h ,0 + k) - f(0,0) = f(h , k) - f(*0,0)
= 0 • h + 0 • k +

•

So that A = 0, B = 0 which does not depends on h and k and

Now approaching along h = mk, we get

which depends on m. Therefore

does not exists. Hence, f(x, y) is not differentiable at origin. Hence, option (b) is incorrect.
Next,

Hence, option (c) is incorrect.
And,

Hence, option (d) is incorrect.

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