A simple integration problem involving step functions.

image containing math problem statement
This simple problem illustrates how to evaluate the definite integral of a function with repetitive pattern.
We will start by analyzing the inner function first. For any value of $x$, $(x - \lfloor x \rfloor)$ lies in the range $[0, 1)$
Therefore, $2 (x - \lfloor x \rfloor)$ is in the range $[0, 2)$

For $x \in [0, 0.5)$ the given function $\displaystyle \lfloor 2\left (x - \lfloor x \rfloor \right) \rfloor = 0$
and for $x \in [0.5, 1)$, $\displaystyle \lfloor 2\left (x - \lfloor x \rfloor \right) \rfloor = 1$

This pattern repeats for any interval $x \in [n, n+1)$ for any integer $n$

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Therefore,
$\displaystyle \int\limits_0^{100} \lfloor 2\left (x - \lfloor x \rfloor \right) \rfloor dx$

$= \displaystyle 100 \int\limits_0^{1} \lfloor 2\left (x - \lfloor x \rfloor \right) \rfloor dx$

$= \displaystyle 100 \left(\int\limits_0^{0.5} \lfloor 2\left (x - \lfloor x \rfloor \right) \rfloor dx + \int\limits_{0.5}^{1} \lfloor 2\left (x - \lfloor x \rfloor \right) \rfloor dx \right)$

$= \displaystyle 100 \left(\int\limits_0^{0.5} 0 dx + \int\limits_{0.5}^{1} 1 dx \right)$

$= \displaystyle 100 \left(0 + 0.5 \right)$

$= 50$