We show generic existence of functions f in the Handy space H p (0 < p < 1) on the open unit disc whose primitive F (f ) satisfies the following. i) F (f ) ∈ H q , where q = p 1 − p .(ii) For every a > q and every A, B ∈ R, A < B it holdsiii) The functions f and F (f ) are totally unbounded and hence non-extendable.Results of similar nature are valid when the space H p is replaced by localized versions of it, 0 < p < 1, or intersections of such spaces.

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