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.

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 $y = -t$.

