unfold T S
⊢ ContinuousOn T (Set.Icc 0 1)
⊢ ContinuousOn
    (fun t =>
      1 / (1 + (1 - t) + t * (1 - t)) + (1 - t) / (1 + (1 - t) + t) + t / (1 - t + t + t * (1 - t)) +
        t * (1 - t) / (1 + t + t * (1 - t)))
    (Set.Icc 0 1)