Is $f(x,y,z) = x+2y-3z$ a continuous function? How do I decide without graphing the function? I am just starting to learn continuity of functions and trying to find out ways how to figure out if a given function is continuous or not. So far I have found that we will have to graph a function to decide if it is continuous or not. is there any other ways to figure out if $f(x,y,z) = x+2y-3z$ a continuous function? 
 A: Yes, $f(x,y,z)=x+2y-3z$ is a continuous function. In fact, any function of the form $f(x,y,z)=ax+by+cz$ for constants $a,b$ and $c$ is continuous. There are a number of rules to help you decide if a given function is continuous. For example, the sum or product of two continuous functions is always continuous. A continuous function raised to a continuous function is always continuous.
Consider the functions $g(x,y)=\sin(x)$ and $h(x,y)=\cos y.$ Because they are both continuous, they can be combined by multiplication, addition, and exponentiation to get new continuous functions. So $h(x,y)=(2\sin x\cos y+\cos^4y)^3$ will also be continuous.
Going back to your example, the functions $x,$ $y$, and $z$ are known to be continuous. So, multiplying by constants, $2y$ and $-3z$ must also be continuous. Then adding these terms,
$$
x+2y-3z
$$
must be continuous too.
A: It depends if you are in first or second year.  For first year, subrosa has a good answer, although you can only raise to a continuous power if the base function is positive.  
For second year, suppose $||(x,y,z)-(x_0,y_0,z_0)||\lt\delta$. Then $|f(x,y,z)-f(x_0,y_0,z_0)|=|(x-x_0)+2(y-y_0)-3(z-z_0)| $. Show it is less than some multiple of the distance between the points, so for any $\epsilon$ you can pick a delta to ensure the f values are within epsilon of each other.
