onsider the function Determine the set of all those points in R2 at which F is continuous. Solution: First note that z? + my + 3/2 : (z+ y)2+ 3/2 : 0 if and only if (m, y) : So, on R2 Without the mint , F is continuous, being a quotient of two polynomials whose denominator does not vanish. Consequently, we are left to investigate F for continuity at the point . To be continuous at , the function F must satisfy limmthiol F(z,y) : F : 0. To determine if there is even a chance that limmwmyo) F(z,y) exists, we compute F(z, y) along lines y : km, and let x ~> 0. We find: F(z, km) : m, km) : 3 (choose one of the following) k (l) 171%}:2 H km W 1+k7k2 (iii) 0, We examine what this means for some sample values of k: - k : 1: limaHo F(z,z) : - k : 71: limgHD F(m7 ,1) : This means that limwthm F(z,y) 3 (choose one of the followmg): (i) Does not eXist (ii): 0 (iii): 1. So: Fis : . Consequently, the set of all those points in R2 at which the function F is continuous is R2 without the point