Figure

The function \(z=e^{t}\) is continuous for all \(t\), and the function \( t=x^{2}+y^{2}\) is continuous on its domain (all points in the plane), so the composite \(z=e^{x^{2}+y^{2}}\) is continuous on its domain (all points in the plane), as shown in Figure 29(a).
The sine function \(z=\sin t\) is continuous for all \(t,\) and the function \(t=e^{x^{2}+y^{2}}\) is continuous for all points \((x,y) \) in the plane, so the composite \(z=\sin e^{x^{2}+y^{2}}\) is continuous for all points \((x,y) \) in the plane, as shown in Figure 29(b).
The function \(z=\ln t\) is continuous for all \(t>0,\) and the function \( t=\dfrac{y}{x}\) is continuous for all \(x\neq 0.\) The composite function \( z=\ln \dfrac{y}{x}\) is continuous for all \((x,y) \) for which \( x>0,\) \(y>0\) or \(x<0,\) \(y<0.\) The graph of \(z\) has no points for which \(x\geq 0,\) \(y\leq 0\) or \(x\leq 0,\) \(y\geq 0\), as shown in Figure 29(c).