Find noncontinuous $F : \Bbb{R}^{\Bbb{R}} \to \Bbb{R}$ such that if $f_n \to f$ in $\Bbb{R}^{\Bbb{R}}$, then $F(f_n) \to F(f)$

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.

The more I look into this, the more I suspect that Willard slipped up here: it appears that the existence of such function depends on the existence of certain large cardinals and hence on one’s set theory. It is possible to find such a function from a subspace of $\mathbb{R}^\mathbb{R}$ to $\mathbb{R}$; one subspace that works is $X = \Sigma_0 \cup \Sigma_1$, where $$\Sigma_0 = \{f \in \{0,1\}^\mathbb{R}:\{x \in \mathbb{R}:f(x) = 0\} \mbox{ is countable}\}$$ and $$\Sigma_1 = \{f \in \{0,1\}^\mathbb{R}:\{x \in \mathbb{R}:f(x) = 1\} \mbox{ is countable}\}.$$ It’s not hard to check that $\Sigma_0$ and $\Sigma_1$ are dense, sequentially closed subsets of $X$. Now just define $F:X \to \mathbb{R}$ by $F(f) = 0$ if $f \in \Sigma_0$ and $F(f) = 1$ if $f \in \Sigma_1$. If $\langle f_n \rangle_n \to f \in \Sigma_i$ (where of course each $f_n \in X$), there is some $n_0$ such that $f_n \in \Sigma_i$ for all $n \ge n_0$, whence $F(f_n) = i = F(f)$ for all $n \ge n_0$, and $\langle F(f_n) \rangle_n \to F(f)$, so $F$ is sequentially continuous. However, $F^{-1}[(-1/2,1/2)] = \Sigma_0$ is not open in $X$, so $F$ isn’t continuous.

If you want to try a substitute exercise based on the material in Section 10, find a sequentially continuous, non-continuous function from $\mathbf{\Omega}$ to $\mathbb{R}$, i.e., a function $f:\mathbf{\Omega} \to \mathbb{R}$ such that $f$ is not continuous, but $\langle f(x_n) \rangle_n \to f(x)$ in $\mathbb{R}$ whenever $\langle x_n \rangle_n \to x$ in $\mathbf{\Omega}$. Hint: This can be done with a function that is constant on the set $\{x \in \mathbf{\Omega}:\omega_0 \le x < \omega_1\}$.

In Willard’s notation $\mathbf\Omega$ denotes the set of ordinals $\{\alpha:1\le \alpha\le\omega_1\}$ with the order topology (i.e. the sets $(\gamma,\omega_1]=\{\alpha\in\mathbf\Omega: \alpha>\gamma\}$, $[1,\gamma)=\{\alpha\in\mathbf\Omega: \alpha<\gamma\}$ for $\gamma\in\mathbf\Omega$ form a subbase).