# Representing rectangular function using divided differences.

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.

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

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

Following this definition I obtain that, assuming (as the autor makes) $$t_j, the first term will be $$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 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?