I was reading the chapter 9 of A practical guide to splines (de Boor, C.) were the autor defines a B-spline of order 1, basically a rectangular function i.e. a function taking value $1$ if the variable $x$ is contained in the interval $(t_j, t_{j+1})$ and $0$ elsewhere using divided differences as follows:

$B_1(x)=(t_{j+1}-t_j)[t_j,t_{j+1}](\cdot -x)^0_+$

where $[t_j,...,t_{j+k}]f$ indicates the k-th divided difference of the function $f$ and $(\cdot)_+$ indicates the positive part of the argument (if the argument is negative then the function returns as result $0$).

This definition is okey to me, basically we have that $B_1(x)=1$ if $t_j<x<t_{j+1}$.

Then the autor says that another way to write it following the definition of divided differences is:


Following this definition I obtain that, assuming (as the autor makes) $t_j<t_{j+1}$, the first term will be $1$ if $x>t_{j+1}$ and given our assumption also the second term will be $1$ leading to a result of $0$. If instead $t_j<x<t_{j+1}$ the first term will be $0$ and the second term will be $1$, but this is a rectangular function with a minus in front.

Am I missing something?

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