Rectangular function

Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way

The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as rect ⁡ ( t T) = Π ( t T) = { 0 , if | t | > T 2 1 2 , if | t | = T 2 1 , if | t | < T 2 . {\displaystyle \operatorname {rect} \left({\frac {t}{T}}\right)=\Pi \left({\frac {t}{T}}\right)=\left\{{\begin{array}{rl}0,&{\text{if }}|t|>{\frac {T}{2}}\\{\frac {1}{2}},&{\text{if }}|t|={\frac {T}{2}}\\1,&{\text{if }}|t|<{\frac {T}{2}}.\end{array}}\right.} Alternative definitions of the function define rect ⁡ ( t = ± T 2) {\textstyle \operatorname {rect} \left(t=\pm {\frac {T}{2}}\right)} to be 0, 1, or undefined. The area under the curve does not change for the different definitions of the functions at t = ± T 2 {\textstyle t=\pm {\frac {T}{2}}} .

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