In the proof, Rudin defines $g(t) = f(t) - P(t) - M(t - \alpha)^n$ and apply Mean Value Theorem (MVT) to $g$ on $[\alpha, \beta] \subset [a,b]$ (assuming $\alpha > \beta$), $g'$ on $[\alpha, x_1]$ and so on.
In order to apply MVT, $g$ needs to be continuous on $[a,b]$, $g'$ continuous on $[a,x_1]$, etc. I am not sure how to check this continuity.
I think there should be a way to establish continuity of $g^{(k)}$ on $[a,b]$ for $k = 0, ..., n-1$. Can some one help on this?
By assumption, $f^{(k)}$ is continuous on $[a,b]$ for $k=0,\dots,n-1$. Since $g$ is just $f$ plus some polynomial (namely $-P(t)-M(t-\alpha)^n$), $g^{(k)}$ is just $f^{(k)}$ plus some polynomial, and therefore is continuous on $[a,b]$.