Answer to Question #121564 in Calculus for Karan

Question

Even if both the partial derivatives of R → R
2
f : exist at a point )b,a( , the
function f may still not be continuous at )b,a( .

Answer 1

#Q121564

Now the following example satisfied the given condition

Exp:

f(x,y)= xy /(x²+y²)

Now let us find partial derivative of f(x,y) with respect to x at (0,0)

f_{x = y / (x}² +y²)

So, f_x(0,0) = 0

_{Now let us find partial derivative of f(x,y) with respect to y at (0,0)}

_{fy = x / (x}²+y²)

So, f_y(0,0)=0

Therefore , both partial derivative of f(x,y) exist at (0,0)

Now let us check continuity of f(x,y) at (0,0)

Put y=mx into f(x,y)

So, f(x,y)= mx²/x².(1+m²) = m/(1+m²)

Hence , limt of f(x,y) as (x,y) $\to$ (0,0)

Depends upon the value of m

So, limit doesn't exist

Hence , f(x,y) is not continuous at (0,0)

Ans. f(x,y)= xy /(x²+y²) satisfied given condition.