Question

discuss the continuity of the function. f(x, y, z) = sin(z) / (e^x + e^y) continuous everywhere continuous except at (0, 0, 0) continuous except where y < 0 continuous except where z < 0 continuous except where x < 0

Explanation:

Step1: Recall continuity condition

A function $f(x,y,z)=\frac{\sin(z)}{e^{x}+e^{y}}$ is continuous where the denominator $e^{x}+e^{y}
eq0$ and $\sin(z)$ is well - defined. Since $e^{x}>0$ and $e^{y}>0$ for all real $x$ and $y$, the denominator is always positive. The function $\sin(z)$ is well - defined for all real $z$. But $\sin(z) = 0$ when $z = n\pi$, $n\in\mathbb{Z}$. The function is not defined when $\sin(z)=0$, i.e., when $z = n\pi$, $n\in\mathbb{Z}$.

Step2: Determine continuity

The exponential functions $e^{x}$ and $e^{y}$ are always positive for real $x$ and $y$. The only issue with the continuity of the given function $f(x,y,z)$ is when the denominator of the fraction is not well - defined due to $\sin(z) = 0$. So the function is continuous everywhere except where $\sin(z)=0$, which is where $z = n\pi,n\in\mathbb{Z}$. In the given options, the correct one is that the function is continuous except where $z<0$ is wrong as the problem is with $z = n\pi$ values. The function is not continuous everywhere. It is not discontinuous only when $x < 0$ or $y < 0$. And it is not discontinuous only at $(0,0,0)$.

Answer:

continuous except where $z = n\pi,n\in\mathbb{Z}$ (but among the given options, the closest correct one is continuous except where $z<0$ is incorrect, continuous everywhere is incorrect, continuous except where $y < 0$ is incorrect, continuous except at $(0,0,0)$ is incorrect. The function is continuous except where $z$ is such that $\sin(z)=0$). If we assume the options are mis - written and we go with the spirit of the problem, the function is continuous except where $z$ takes values for which $\sin(z) = 0$. If we must choose from the given options, there is no completely correct option, but the most relevant incorrect - option analysis shows that the function has issues related to $z$ values and not $x$ or $y$ values or just the origin).