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Continuous triangular spikes on converge pointwise to zero but not uniformly when monotonicity is absent
Statement refuted
Refuted claim: the monotonicity hypothesis in Dini's theorem can be dropped.
For put and define the triangular spike
Each is continuous and pointwise, but the convergence is not uniform.
Facts & Assumptions
Given: The functions in the Statement, with .
Constants, the identity, sums, products, absolute values, and pointwise maxima of continuous real functions are 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, Maximum and minimum of a set, Continuity of at a point of and on : the - condition, its agreement with at a limit point, and continuity at an isolated point).
For every real there is with ; canonical naturals increase and positive reciprocals decrease (For every in a complete ordered field there is a natural with , The canonical natural of a field, Canonical naturals are positive and strictly increasing, Inverses of positives are positive, and reciprocation reverses order).
Absolute value is nonnegative and has the usual multiplicative law (Basic properties of the absolute value).
Dini's theorem on a closed interval requires one pointwise monotonicity direction for the whole sequence (Dini's theorem on a closed interval: monotone pointwise convergence of continuous functions to a continuous limit is uniform).
Counterexample
Every is continuous by [L1], and the zero function is continuous.
If , then . If , choose with ; for all sufficiently large , , so and . Thus pointwise.
At one has , so the convergence is not uniform.
At , the values at are respectively , so the sequence is neither pointwise nondecreasing nor pointwise nonincreasing.
All Dini hypotheses except monotonicity hold while uniform convergence fails, so monotonicity cannot be dropped.
Depends on
- Dini's theorem on a closed interval: monotone pointwise convergence of continuous functions to a continuous limit is uniform
- 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
- Maximum and minimum of a set
- For every $\varepsilon > 0$ in a complete ordered field there is a natural $n \ge 1$ with $1/n < \varepsilon$
- The canonical natural $\iota(n) = n \cdot 1_F$ of a field
- Canonical naturals are positive and strictly increasing
- Inverses of positives are positive, and reciprocation reverses order
- Basic properties of the absolute value
Used by
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Sources
- Dini's theorem (Wikipedia) (standard reference, not scraped)
- W. Trench, Introduction to Real Analysis (standard reference, not scraped)