1. ## Peicewise Functions Continuity

I am doing some review, starting back at the fundamentals and I do not know why but piecewise confuse the hell out of me at times. I have the following piecewise function and have to tell if it is continuous or not on the interval [-1,1]

$\displaystyle f(x)=\frac{x}{|x|} x not equal to 0$

$\displaystyle f(x)= x = 0 when x = 0$

Sorry to be so abrupt, I know how to tell continuity with normal functions such as if it a rational fraction and long as the value that deems the function undefined is not in that interval it is continuous? also if the sign changes from left to right in a interval?

Any tips to help me with continuity

2. Originally Posted by The Power
Any tips to help me with continuity
Graph it out and make sure it doesn't have any gaps

3. We are told to evaluate it algebraically since we are not allowed to use calculators on a test. Thanks for the help though, it know its not terrible hard, its just that piecewise functions to through me off at times.

Edit: I think I finally recall my information and remove this mental block. A piecewise function is a whole function just has different rules for set intervals. So in sense if the interval for
$\displaystyle f(x)=\frac{x}{|x|}$ when x is not equal to 0,

Sorry I forgot latex tags. So if the rule was a tad different and allowed 0, this would cause the function to be undefined causing a &quot;gap&quot; or hole in the graph thus rendering it no longer continuous, however since we have the rule when x equals 0 the output is 0 this allow the graph to continue on the interval [-1,1] Please someone correct me if I am wrong.

4. $\displaystyle \frac{x}{{\left| x \right|}} = \left\{ {\begin{array}{rl} {1,} & {x > 0} \\ { - 1,} & {x < 0} \\ \end{array} } \right.$

5. As Plato said, if x< 0, |x|= -x so $\displaystyle \frac{|x|}{x}= -1$ for x< 1. If x> 0, |x|= x so $\displaystyle \frac{|x|}{x}= 1$ for x> 1.

In order for the function f(x) to be continuous at x= a, three things must be true:
1) f(a) is defined
2) $\displaystyle \lim_{x\rightarrow a} f(x)$ exists
3) $\displaystyle \lim_{x\rightarrow a} f(x)= f(a)$
(Since $\displaystyle \lim_{x\rightarrow a} f(x)= f(a)$ pretty much implies the two sides exist, usuallly we just state (3).)

Now, if $\displaystyle \lim_{x\rightarrow a} f(x)$ exists then the two "onesided limits", $\displaystyle \lim_{x\rightarrow a^-} f(x)$ and $\displaystyle \lim_{x\rightarrow a^+} f(x)$ must exist and be equal. Think about what that means here.