Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p\_{f,A,n}(x)$ of degree $n$ such that $f(a)=p\_{f,A,n}(a),\forall a\in A$. Such a polynomial is called the interpolating polynomial of $f$ with nodes $A$. Due to this, given $A\subseteq [a,b]$ with $n+1=|A|$, we can consider a linear map $X\_{A,n} : C[a,b]\to C[a,b]$ such that $X\_{A,n}(f)=p\_{A,f,n}$ . Then $X\_{A,n}$ is a continuous map and also a projection i.e. $X^2\_{A,n}=X\_{A,n}$. It can be shown that there is a constant $c$ such that $||X\_{A,n}||\_{\infty} \ge c \ln (n)$ for any set $A\subseteq [a,b]$ of size $n+1$, where $||.||\_{\infty}$ is the $L^{\infty}$ operator norm induced from the $L^{\infty}$ norm on $C[a,b]$. Since $(C[a,b],||.||\_{\infty})$ is a Banach space, it follows from Uniform boundedness principle, that given any sequence of subsets $\{A\_n\}\_{n=1}^\infty$ of $[a,b]$ with each $A\_n$ having size $n+1$, $\exists f\in C[a,b]$ such that $||X\_{A\_n,n}(f) - f||\_{\infty} $ does not converge to $0$.


This can be rephrased as: Given any sequence $\{A\_n\}\_{n=1}^\infty$ of subsets of $[a,b]$, with each $A\_n$ of size $n+1$, there exists a continuous function $f$ on $[a,b]$ such that the sequence of interpolating polynomials $p\_{f,A\_n,n}(x)$ doesn't converge to $f$ uniformly on $C[a,b]$. 


**My question** is the following: Given any sequence $\{A\_n\}\_{n=1}^\infty$ of subsets of $[a,b]$, with each $A\_n$ of size $n+1$, does there exist a continuous function $f$ on $[a,b]$ such that the sequence of interpolating polynomials $p\_{f,A\_n,n}(x)$ doesn't converge to $f$ point-wise at atleast one point ? 



Given any sequence of interpolating nodes, can we find a continuous function $f$ whose interpolating polynomials doesn't converge to $f$ point-wise
The result that you mention in the first part of your question is a classical result by Faber



> 
> G. Faber, Uber die interpolatorsche Darstellung stetiger Funktionen, Jahresber. der deutschen Math. Verein. 23 (1914), 190-210.
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Of course, this result does not exclude pointwise convergence. This question was negatively
answered by S. Bernstein in



> 
> S. Bernstein, [Quelques remarques sur l'interpolation](https://eudml.org/doc/158778), Math. Ann. 79
>  (1918), 1-12; doi: [10.1007/BF01457173](https://doi.org/10.1007/BF01457173).
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For an arbitrary scheme $X=\cup\_{n} A\_{n}$ of points in $[-1,1]$, there exist a continuous function $f$ and a point $x\_{0}$ in $[-1,1]$ such that
$$
 { \limsup \_ { n \rightarrow \infty } } \left| L \_ { n } \left( f, X , x \_ { 0 } \right) \right| = \infty.
$$
The next natural question is the possibility of divergence on a set of positive measure. Such a result was obtained by Marcinkiewicz and Grunwald (independently) for the particular scheme $T$ of Chebyshev nodes :


There exists a function continuous $f$ such that
$$
{ \limsup \_ { n \rightarrow \infty } } \left| L \_ { n } \left( f, T , x \right) \right| = \infty \quad \text { for all } \quad x \in [ - 1,1 ].
$$
There is an explicit construction of such a function in the book by I.P. Natanson, Constructive Function Theory, Vol. 3, pp. 35-46.


P. Erdos made the conjecture that this negative result holds for an arbitrary scheme $X$ of points in $[-1,1]$. This was proved in



> 
> P. Erdos and P. Vertesi, [On the almost everywhere divergence of
>  Lagrange interpolatory polynomials for arbitrary system of nodes](https://users.renyi.hu/~p_erdos/1981-03.pdf), Acta
>  Math. Acad. Sci. Hungar. 36 (1980), 71-89 and 38 (1981), 263; doi: [10.1007/BF01897094](https://doi.org/10.1007/BF01897094).
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For any scheme of points $X$ in $[-1,1]$, there exists a function $f$ such that
$$
{ \limsup \_ { n \rightarrow \infty } } \left| L \_ { n } \left( f, T , x \right) \right| = \infty \quad \text {almost everywhere in} \quad x \in [ - 1,1 ],
$$
and the divergence set is of second category. Note that divergence everywhere is not possible just by considering a newtonian scheme i.e. a scheme that repeats points ($A\_{n}\subset A\_{n+1}$, $n\geq1$, in your notation).


A very nice book about classical interpolation of functions is



> 
> Szabados, J., Vértesi, P., Interpolation of functions. World
>  Scientific Publishing Co., Teaneck, NJ, 1990.
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