TITLE: Find noncontinuous $F : \Bbb{R}^{\Bbb{R}} \to \Bbb{R}$ such that if $f_n \to f$ in $\Bbb{R}^{\Bbb{R}}$, then $F(f_n) \to F(f)$
QUESTION [18 upvotes]: I'm reading the book General Topology by S. Willard and I came across the following problem.
We have to find an example of a noncontinuous function $F:\mathbb R^\mathbb R\to \mathbb R$ with the property that
whenever $f_n\to f$ in $\mathbb R^\mathbb R$, then $F(f_n)\to F(f).$ Here $\mathbb R^\mathbb R$ denote a space 
with the product topology and $\mathbb R$ is equipped with the usual topology. Please give
me some hint.
Edit (T.B.) The problem comes from section 10, specifically as a problem illustrating the points in (10.6).

REPLY [7 votes]: It is not an answer, but this example may be funny.
Let $X$ be the subspace of $]0,1[^{]0,1[}$ consisting of measurable functions (with the product topology), and let $F : X \rightarrow ]0,1[, f \mapsto \int f$.
Then $F$ is sequentially continuous by the Dominated Convergence Theorem, but nowhere continuous since it is surjective on every open subset.