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).