# Complex Variables: Continuity

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

Krayfish
Mark44
Mentor
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...

Krayfish
RPinPA
Homework Helper
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")

Krayfish
vela
Staff Emeritus
Homework Helper

## 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