discuss the continuity of the function. f(x, y, z) = sin(z) / (e^x + e^y) continuous everywhere continuous…

Explanation:

Step1: Recall continuity condition

A function (f(x,y,z)=\frac{\sin(z)}{e^{x}+e^{y}}) is continuous at a point ((a,b,c)) if (\lim_{(x,y,z)\to(a,b,c)}f(x,y,z)=f(a,b,c)) and the function is well - defined at ((a,b,c)). The exponential functions (y = e^{x}) and (y=e^{y}) are defined for all real (x) and (y), and the sine function (y = \sin(z)) is defined for all real (z). However, the denominator (e^{x}+e^{y}\neq0) for all real (x) and (y) since (e^{x}>0) and (e^{y}>0) for all (x,y\in R). The function (f(x,y,z)) is undefined when (\sin(z) = 0), i.e., (z = k\pi), (k\in\mathbb{Z}).

Step2: Analyze the domain of continuity

The function (f(x,y,z)) is a quotient of two continuous functions. The numerator (g(z)=\sin(z)) and the denominator (h(x,y)=e^{x}+e^{y}). The function (f(x,y,z)) is continuous everywhere except where (\sin(z)=0), or (z = k\pi,k\in\mathbb{Z}). When (z = k\pi), the denominator (e^{x}+e^{y}\neq0) for all real (x) and (y), but the function is not well - defined due to the zero in the numerator.

Answer:

continuous except where (z = k\pi,k\in\mathbb{Z}) (none of the given options are correct as stated in the problem - if we assume the closest is the option about (z) values, we can say the function is continuous except where (z<0) is a wrong statement, continuous everywhere is wrong, continuous except where (y < 0) is wrong, continuous except at ((0,0,0)) is wrong. The correct answer should be continuous except where (z) is an integer multiple of (\pi))