Recently, I'm reading a paper Riesz representation theorem, Borel measures and subsystems of second-order arithmetic. In page 2, the author said that polynomials are linear combinations of basic functions with rational coefficients, while the basic functions are defined as: $p(x) = \begin{cases} 1 ~~~~~~~~~~~~,~ d(a,x)\leq s \\ \frac{r-d(a,x)}{r-s}~,~ s < d(a,x) < r \\ 0 ~~~~~~~~~~~~,~ r \leq d(a,x) \end{cases} $
I think the graph of the basic function mentioned above is like a truncated cone, while some points of the basic function are not differentiable, so I'm confused that how a polynomial can be represented as a linear combination of basic functions. Any help will be appreciated.