p-points in topology A point $x_\infty$ in a topological space $X$ is called a p-point if every continuous function on $X$ is constant in  a neighborhood 
  of $x_\infty$.
  For example if $X=X_0 \cup x_\infty$ where $X_0$ is an uncountable discrete space and the neighborhoods of $x_\infty$ are co-countable, 
  then $x_\infty$ is such a p-point for $X$. Also,
  the "corona"  $\beta\mathbb N\setminus \mathbb N$ in the Stone-Cech compactification of the integers admits p-points under the continuum hypothesis.
  Can one prove in an elementary way (as in the first example above  without using the concept of ordinals)  the existence of compact Hausdorff spaces  that admit non-isolate p-points?

$X=\omega_1 +1$ in the order topology has a non-isolated $p$-point in your sense (namely $\omega_1$) and is compact Hausdorff. 
This is well-known: suppose $f: \omega+1 \to Y$ is continuous and $Y$ is first countable (having countable pseudocharacter will also do). Then let $p = f(\omega_1) \in Y$ and let $U_n$ be a countable neighbourhood base at $p$. Then for each $n$ pick $\alpha_n < \omega_1$ such that $f[(\alpha_n, \omega_1]] \subseteq U_n$ by continuity of $f$ a $\omega_1$, and define $\beta = \sup_n \alpha_n < \omega_1$, then $f$ is constantly $p$ on the neighbourhood $(\beta, \omega_1]$ of $\omega_1$.
Don't confuse your notion with the much stronger notion of $P$-point that is used for points in the Stone-Cech remainder of $\mathbb{N}$; these need not exist, as Shelah has shown, but I think there probably are $p$-points in your sense in that remainder, maybe Kunen's weak $P$-points (see this paper and its references for more info on those) will do it.
