Need help with the question

the options for the fifth blank are rational function, irrational function, or polynomial

Let my F(,y] : { w2+wy+y2 Determine the largest set of points at which the function is continuous Solution: First note that the domain of the function fb, y) = % is R2 Without the point w w y since :2 + W + y? : (x+ yj2+ y? ? 0 if and only if (my) 5A Since 3' is E the function f is continuous on its domain. This means that the function F is continuous on R2 Without the point Hencer there is only one paint left to investigate continuity, the point .To be continuous at limwwamm F(3,y) = F By the defintion ofF,F(0,0) = 0. To find llmKwHJKUD) F(, y) , let (my) 4) (0,0) along lines 3; = fez. Since F($, k3) = f(z, Fax) = 3 (choose one of the following) . k l 0 1+J+J2 lea: 1+J+J (iii) 0, (ii) A we obtain that llm(x,y)i(0,j F(z, g) V (choose one of the following): (i) DIVE (ii) 0 (ill) 1. Hence Fis 3 . Finally, the largest set of points at which the function F is continuous is R2 without the point Way) 7s (0)0) if (my) : (0,0) , the function F must satisfy