Proof that interpolation converges; Reference request I am interested in the mathematical justification for methods of approximating functions.
In $x \in (C[a, b], ||\cdot||_{\infty})$ we know that we can get an arbitrarily good approximation by using high enough order polynomials (Weierstrass Theorem).
Suppose that $x \in (C[a, b], ||\cdot||_{\infty})$. Let $y_n$ be defined by linearly interpolating $x$ on an uniform partition of $[a, b]$ (equidistant nodes). Is it true that
\begin{equation}
  \lim_{n \to \infty} ||y_n - x||_{\infty} = 0?
\end{equation}
Do we need to impose stronger conditions? For example
\begin{equation}
  x(t) = 
  \begin{cases}
    t \sin\left(\frac{1}{t}\right), & t \in (0, \pi] \\
    0,           & t = 0
  \end{cases}
\end{equation}
is in $C[0, 1]$, however it seems to me that we cannot get a good approximation near $t = 0$.
More generally, can anyone recommend a reference containing the theory of linear interpolation and splines? It would have to include conditions under which these approximation methods converge (in some metric) to the true function.
 A: Given an arbitrary function in $x \in C[a, b]$ and defining $y_n$ to be the linear interpolant on the uniform partition of $[a, b]$ with $n + 1$ nodes we have
\begin{equation}
  \lim_{n \to \infty} ||y_n - x||_{\infty} = 0.
\end{equation}
Proof. As $x$ is continuous on the compact set $[a, b]$ it is uniformly continuous. Fix $\varepsilon > 0$. By uniform continuity there exists $\delta > 0$ such that for all $r, s \in [0, 1]$ we have
\begin{equation}
  |r - s| < \delta \quad \Rightarrow \quad |x(r) - x(s)| < \varepsilon.
\end{equation}
Every $n \in \mathbb{N}$ defines a unique uniform partition of $[a, b]$ into $a = t_0 < \ldots < t_n = b$ where $\Delta t_n = t_{l+1} - t_l = t_{k+1} - t_k$ for all $l, k \in \{0, \ldots, n\}$. Choose $N \in \mathbb{N}$ so that $\Delta t_N < \delta$. Let $I_k = [t_k, t_{k+1}]$, $\,k \in \{1, \ldots, N\}$. Then for all $t \in I_k$ we have
\begin{equation}
  |y_N(t) - x(t)| \leq |y_N(t_k) - x(t)| + |y_N(t_{k+1}) - x(t)| < 2 \varepsilon,
\end{equation}
where the first inequality is due to the fact that since $y_N$ is linear on $I_k$ we know that $y_N(t) \in [\min(y_N(t_k), y_N(t_{k+1}), \max(y_N(t_k), y_N(t_{k+1})]$.
Q.E.D.
If anyone knows a reference for a proof along these lines, then I would be grateful to know it.
Also, the function $x$ in the OP can certainly be well approximated near zero. Here is a picture of the function; the dashed lines are $y = t$ and $y = -t$.

