Vtyjkyuk

if $f(x)=int_0^x lfloortrfloor ,dt$ for $x≥0$, draw the graph of
$f$ over the interval [0,4]

This is an exercise from Apostol's Calculus Vol.1.

I quite don't get how to write $int_0^x lfloortrfloor ,dt$ as an algebraic expression without knowing calculus' fundamental theorem and $lfloortrfloor$'s derivative, which haven't been introduced yet.

I input the function in a function plotter, and came up with this.

But I need to know why the graph is like this, to do the problem for my self, and understand it.

Could you help me?

Thanks in advance

If $x in [0,1)$, $f(x) = int_0^x lfloor t rfloor , dt= int_0^x 0, dt=0$.

If $x in [1,2)$, $f(x) = int_0^1 lfloor trfloor , dt + int_1^x lfloor t rfloor , dt= int_0^1 0, dt + int_1^x 1, dt=x-1$.

If $x in [2, 3)$, $f(x) = int_0^1 lfloor trfloor , dt + int_1^2 lfloor trfloor , dt + int_2^x lfloor t rfloor , dt= int_0^1 0, dt + int_1^2 1, dt + int_2^x 2, dt=1+2(x-2)$.

In general, if $x in [n, n+1), n in mathbbN cup 0$,

beginalign
f(x) &= sum_i=0^n-1 int_i^i+1 lfloor t rfloor , dt + int_n^x lfloor t rfloor , dt \
&= sum_i=0^n-1 int_i^i+1 i , dt + int_n^x n , dt \
&= left(sum_i=0^n-1 i right)+ n(x-n)
endalign

Try to simplify the last term.