This applet shows a few piecewise defined functions. Clicking next will advance through these options.[br][list][br][*] [math]sgn(x)[/math] is the sign of [math]x[/math] , [math]y\:=\left\lbrace \begin{array}{cl} 1&\text{: if }x>0\\ -1&\text{: if }x<0\\ 0&\text{: if }x=0\end{array}\right .[/math][br][/*][*] A left function and a right function where you can define both functions and move the split point[br][/*][*] [math]\lfloor x \rfloor[/math] is [math]\text{floor}(x)[/math] which rounds [math]x[/math] down to the next integer[br][/*][*] [math]\lceil x \rceil[/math] is [math]\text{ceil}(x)[/math] which rounds [math]x[/math] up to the next integer[br][/*][*] A function with 3 piecewise components [br][/*][/list][br]Clicking the next button will advance to the next function.

For each function move the [math]x[/math] value and notice what happens with the [math]y[/math] value.[br]What happens when [math]x[/math] is located at a hollow circle?[br]For the [math]\sin ( x ) [/math] and [math]\cos ( x ) [/math] split function what happens at [math]x = \frac{\pi}{4} \approx 0.785 [/math] ?[br]What is the significance of filled and closed circles?