Is there any well defined $f:\mathbb{R}\to\mathbb{R}$, that is not the limit of any sequence of polynomials? Can all functions, $f:\mathbb{R}\to\mathbb{R}$ be given as the limit of a sequence of polynomials $\left(p_i\right)_{i\in\mathbb{N}}$ with real coefficients? Is there a particular function that categorically cannot? The convergence may be any type, pointwise, uniform, etc. and the function may be as discontinuous and pathological as you want.
Of course, well-behaved elementary functions like $\sin x$ and $\exp x$ can be given as Taylor Series while discontinuous functions such as $\mathrm{sgn} x$ and $\lfloor x \rfloor$ can be given as Fourier Series, which in turn can be formed of Taylor Series but is there some exception?
I think this should be a relatively clear-cut question and I've got a feeling the answer is an obvious yes but I really don't know.
Regards, Jam.
 A: Yes. Here is a nonconstructive proof: There are $|\mathbb R|^{|\mathbb R|}=\left(2^{\aleph_0}\right)^{\left(2^{\aleph_0}\right)}=2^{2^{\aleph_0}}$ functions $f : \mathbb R \rightarrow \mathbb R$. However, there are only $|\mathbb R|$ finite tuples of real numbers, hence only $|\mathbb R|=2^{\aleph_0}$ polynomials. 
That means that there are only $|\mathbb R|^{|\mathbb N|}=\left(2^{\aleph_0}\right)^{\aleph_0}=2^{\aleph_0}$ sequences of polynomials. Hence there are at most $2^{\aleph_0}$ functions $f : \mathbb R \rightarrow \mathbb R$ that can be written as the limit of a sequence of polynomials with real coefficients. 
Since $2^{2^{\aleph_0}}>2^{\aleph_0}$ by Cantor's theorem, there is at least one function $f : \mathbb R \rightarrow \mathbb R$ that cannot be written as the limit of a sequence of polynomials with real coefficients. In fact, almost all functions can't be written as the limit of a sequence of polynomials with real coefficients.
A: One thing to note is that $f$ is the pointwise limit of a sequence of polynomials iff $f$ is the pointwise limit of a sequence of continuous functions. This follows from the Weierstrass approximation theorem.
Another thing to note is that by Baire, every pointwise limit of a sequence of continuous functions on $\mathbb R$ is continuous on a dense $G_{\delta}$ subset of $\mathbb R.$ Thus $\chi_{\mathbb Q}$ is a counterexample.
A: Probably not the answer you are looking for, but remember that when dealing with various types of convergence you should work with nets not sequences.
Let $f : \mathbb R \to \mathbb R$ be any function.
Define $A$ to be the set of all finite subsets of $\mathbb R$. For each $\alpha \in A$ define $p_\alpha$ to be any polynomial which coincides with $f(x)$ on $\alpha$ [you can use Lagrange interpolation here]. 
Then $(p_\alpha)_\alpha$ is a net of polynomials, which converges on finite sets (hence poinwise) to $f(x)$. 
