Consider `f(x,y)=(xy)/(x^2+y^2)` if `(x,y) ne (0,0) and f (0,0) = 0` . Show that f is not continuous at (0,0) and continuous at all other points of `R^2.`

Let F (x,y) = (2x - 3y +4)/(x ^(2) + y ^(2) +4) for all (x,y) in R ^(2), Show that f is continuous on R ^(2).

If f (x,y) =e ^(2x+3y) find 3f_(x) (0,0) - 2f_(y) (0,0):

If f(x,y) = 2x^(2) - 3xy + 5y^(2) +7 , then f(0,0) and f(1,1) is

Let f(x,y) = (y^(2) -xy)/(sqrt(x)-sqrt(y)) for (x,y) ne (0,0) . Show that lim_((x,y) to (0,0)) f(x,y) =0

Let g(x,y) = (x^(2)y)/(x^(4) + y^(2)) for (x,y) ne (0,0) and f(0,0)=0. (i) Show that lim_((x,y) to (0,0)) g(x,y)=0 along every line y=mx, m in R . (ii) Show that lim_((x,y) to (0,0)) g(x,y) = k/(1+k^(2)) , along every parabola y=kx^(2), k in R{0} .

Let f(x+y)=f(x)+f(y) for all xa n dydot If the function f(x) is continuous at x=0, show that f(x) is continuous for all xdot

if f(x)={x^3 x<0 3a+x^2 x≥0 is continuous at x = 0 , then a is