Problems of the week (optional)

Show that for any a<b,

B[a,...,a,b](x) = (b-x)^d / (b-a)^d B[a,b](x)

and

B[a,b,...,b](x) = (x-a)^d / (b-a)^d B[a,b](x),

where a and b are repeated d+1 times respectively. Use this to show that 

B[a,b,...,b,c](x) = (x-a)^d / (b-a)^d B[a,b](x) + (c-x)^d / (c-b)^d B[b,c](x)

where b is repeated d times, and show that this function is continuous for all x.