Consider the (complex) Stone-Weierstrass Theorem:
Let $f \in C([a,b], \mathbb{C})$. Then for every $ \epsilon >0$ there exists a polynomial (with complex coefficients) such that $$ \sup_{t \in [a,b]} \vert f(t) - P(t)\vert < \epsilon. $$
From this as a corollary we also get that if $f$ is real-valued, we can find a polynomial with real coefficients for instance $\Re P$ where $P$ is taken from the theorem above.
My question is that in this real setting if we are given a $c \in [a,b]$, whether or not there is a polynomial $P$ such that $f(c)=P(c)$?
My intuition from sketching tells me yes but I am not sure how to prove it. My first instinct was to use the intermediate value theorem and I have also tried manually to tinker with $P$ such that the constant term is $f(c)$. But of these approaches have been unfruitful so far.
Can anyone help me?
