Q: Suppose that the oscillation ω_f (x) of a function f is smaller than η at each point x of an interval [c,d]. Show that there must be a partition π of [c,d] s.t. the oscillation ωf([x_(k-1),x_k ])<η on each member of the partition. My solution (Rough sketch): This condition on x is local, so it must be true for a δ-neightborhood of x s.t. ωf(δ(x))<η. Now take a partition s.t. each subinterval [x_(k-1),x_k ]<δ. Thus, each subinterval is less than the δ from the δ-neightborhood of x, so then ωf([x_(k-1),x_k ])[itex]\leq[/itex]ωf(δ(x))<η. QED Is this logic too sloppy? If so, does anyone have any suggestions as to a more proper way to prove this?