Is f(x) = (x-iy)/(x-1) a Continuous Function?

[Krayfish]

Homework Statement

Determine if the following function is continuous: f(x) = (x-iy)/(x-1)

Homework Equations

How do find out if a function is continuous without graphing it and without a point to examine? I know I've learned this, probably in pre-calculus too, but I'm blanking

The Attempt at a Solution

u(x) = x-iy as a function is continuous because, due to the i term, x-iy will never equal 0 and it is a linear function
v(x) = x-1 is also continuous,as it is a linear function that exists under all conditions

if u and v are both continuous under all conditions, than u/v must also be continuous?

 

[verty]

From the point of view of precalculus, you have learned about limits, in particular, one-sided limits. What happens at x = 1?

 

[Mark44]

if u and v are both continuous under all conditions, than u/v must also be continuous?

u/v will be continuous at points for which v ≠ 0...

 

[RPinPA]

Informally, you can't divide by 0. The function may blow up to +-infinity there. At any rate, it's not defined when the denominator is 0.

Formally, you should see a theorem in your book that for f(x) and g(x) continuous, f(x)/g(x) is continuous except where g(x) = 0. The limit of f(x)/g(x) may exist as g(x)->0, for instance in the function sin(x) / x, but the function itself is not continuous there. (This is a fine point called a "removable discontinuity", which is also in your textbook. Basically, if the limit exists, then you define the function to say "and when the denominator = 0, it's defined to be the limit")

 

[vela]

Homework Statement

Determine if the following function is continuous: f(x) = (x-iy)/(x-1)

Did you mean $f(z)$? If not, then how is $y$ defined?

u(x) = x-iy as a function is continuous because, due to the i term, x-iy will never equal 0 and it is a linear function

Does it matter if $u$ is ever equal to 0?

 

[Krayfish]

From the point of view of precalculus, you have learned about limits, in particular, one-sided limits. What happens at x = 1?

OH yeah that one was pretty obvious, that's what you get when you're doing math with no sleep. Undefined, thank you