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Dini's theorem fails for a discontinuous limit: powers on decrease pointwise to a discontinuous endpoint indicator but not uniformly
Statement refuted
Refuted claim: continuity of the pointwise limit in Dini's theorem can be dropped.
On let . These continuous functions decrease pointwise to the discontinuous endpoint indicator
and the convergence is not uniform.
Facts & Assumptions
Given: The functions and the endpoint indicator on .
The powers converge pointwise to on and do not converge uniformly there ( converges pointwise but not uniformly on ).
Every polynomial function, hence every natural power, is continuous (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, Integer powers , Continuity of at a point of and on : the - condition, its agreement with at a limit point, and continuity at an isolated point).
Dini's theorem on a closed interval requires the approximating functions and their pointwise limit to be continuous (Dini's theorem on a closed interval: monotone pointwise convergence of continuous functions to a continuous limit is uniform).
Continuity at requires that every positive output error admit a positive input radius on which all nearby function values remain close to the value at (Continuity of at a point of and on : the - condition, its agreement with at a limit point, and continuity at an isolated point).
Counterexample
Each is continuous by [L2].
For , , so the sequence is pointwise nonincreasing.
The pointwise convergence to and the failure of uniform convergence are [L1].
The function is discontinuous at : for every , the point lies in with and .
Thus compactness, continuity of all approximants, and monotonicity hold, but the limit is discontinuous and the uniform conclusion fails; continuity of the limit in [L3] is indispensable.
Depends on
- Dini's theorem on a closed interval: monotone pointwise convergence of continuous functions to a continuous limit is uniform
- $f_k(x)=x^{k+1}$ converges pointwise but not uniformly on $[0,1]$
- Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point
- Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function
- Integer powers $a^m$
Used by
Nothing in the library uses this result yet.
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Sources
- J. Lebl, Basic Analysis I, §6.1 (standard reference, not scraped)
- Dini's theorem (Wikipedia) (standard reference, not scraped)
- W. Trench, Introduction to Real Analysis (standard reference, not scraped)