# What kind of proof should I be creating for the C^p-1 continuity of B-Splines?

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