# Continuity on piecewise function

1. Jul 1, 2011

### Wables

1. The problem statement, all variables and given/known data
[10 Marks] At which points is the following function continuous and at which point is it discontinuous. Explain the types of discontinuity at each point where the function is discontinuous. Then at each point of the discontinuity, if possible, find a value for f(x) that makes it continuous or one sided continuous.

f(x) =
-2x if -1$\leq$x<1
-2/(x-1) if 1<x<2
x-2 if x> 2
2. Relevant equations

Test continuity at point:
f(a) is defined
lim f(x) exists
x->a
lim f(x) = f(a)
x->a

Continuity at Endpoints
lim f(x) = f(a) = left continuous
x->a-
lim f(x) = f(a) = right continuous
x->a+

3. The attempt at a solution

Im thinking what I need to do, is:
Check for continuity at the points -1, 1, 2.
Then I would classify any discontinuities as either removable, jump, or infinite discontinuities.
But that last part, im not sure what its asking?
What I have is this so far:

Discontinuous at x=1 and x=2.
At x=1: Jump discontinuity
At x=2: Jump discontinuity

Im not sure if im supposed to do this or if its right, but I did it anyway:
At x=-1, the function is right continuous on the interval [-1, 1). The function is also continuous on (1, 2) and (2, infinity)

2. Jul 1, 2011

### HallsofIvy

Staff Emeritus
Basically correct but the discontinuity at x= 1 is NOT a "jump" discontinuity:
$$\lim_{x\to 1^+} f(x)= \lim_{x\to 1}\frac{-2}{x- 1}= -\infty$$

3. Jul 1, 2011

### Wables

Oh really? Cool! Thanks! So by stating the intervals of continuity, I satisfied the last part of the problem? Cause I was not sure what it was asking..