Suppose u have a function $$f \colon (S^2) ^4\to \mathbb{R} $$ such that $f(a, b, c, d) = f(\sigma(a, b, c, d)) $ for all permutation $\sigma \in \mathcal{S}_4$ and such that $f$ is continuos if restricted to the first three component (i mean, for all $x_0 \in S^2$ the restriction $f_{|(S^2)^3 \times \{x_0\}} $ is continuos).
Is then $f$ continuos?
Thanks in advice.