Determining Where a Function Is Continuous

Determine where the following functions are continuous:

1. \(f(x,y) =3xy^{2}-5xy+4x^{3}y\)
2. \( R(x,y) =\dfrac{x^{2}y-2xy}{xy-1}\)

Solution (a) Since \(f\) is a polynomial function, it is continuous on its domain, every point \((x,y) \) in the plane, as shown in Figure 28(a).

(b) \(R\) is a rational function and is continuous on its domain, all points \((x,y) \) in the plane, except those for which \(xy=1.\) The graph of \(R\) will have no points on the cylinder \(xy=1\), as shown in Figure 28(b).