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Reverted good faith edits by 138.118.49.43 (talk): ">" should not be changed to "<", as the next paragraph says "maps outside the ε-neighborhood" |
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Fix an arbitrary ''ε'' > 0. Then for any ''δ'' > 0 consider the set ''B<sub>δ</sub>'' defined as
: <math>
B_\delta = \big\{x\in S \mid x\notin D_g:\ \exists y\in S:\ |x-y|<\delta,\, |g(x)-g(y)|
</math>
This is the set of continuity points ''x'' of the function ''g''(·) for which it is possible to find, within the ''δ''-neighborhood of ''x'', a point which maps outside the ''ε''-neighborhood of ''g''(''x''). By definition of continuity, this set shrinks as ''δ'' goes to zero, so that lim<sub>''δ'' → 0</sub>''B<sub>δ</sub>'' = ∅.
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