So for a continuous section f , the real axis of Heath’s V-space is the join of all the various sets V_i, n , and by the Baire Category Theorem these cannot all be nowhere dense “in the reals”. So at least one V_i, n is dense in some non-trivial real interval.
  In such a V_i, n we can therefore find two x-axis points that are closer together than 2/n , and for such points, their basic open Vs of finger-height 1/n ...

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