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So I'm working on my project and trying to prove this theorem:

$B_{j,p,t}(x)$ has $C^{p-1}$ continuity at each simple knot where $p\geqslant1$ is the degree of the polynomial.

My first idea was to use proof by induction and I'm just getting stuck on the inductive step.

Like I understand why this is true as any $B_{j,p,t}(x)$ is gonna be at least a function of degree p-1 so you can easily take the p-1 derivative but I just have no idea how to prove that

$B_{j,p+1,t}(x)=\frac{x-t_{j}}{t_{j+p+1}-t_{j}}B_{j,p,t}(x)+\frac{t_{j+2+p}-x}{t_{j+2+p}-t_{j+1}}B_{j+1,p,t}(x) $

is a $C^{p}$ continuous function

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