A note on feebly continuous functions

A function f from R 2 to R is said to be feebly continuous at a point ( x , y ) if there exist sequences x n ↘ x and y n ↘ y with lim n → ∞ lim m → ∞ f ( x n , y m ) = f ( x , y ) . Dales asked if every function has a point of feeble continuity. Our aim in this short note is to show that (assuming the Continuum Hypothesis) the answer is ‘no’. Dales also asked what happens for functions taking only two values: we show that in this case the answer is ‘yes’.