به انگلیسی

One delta for every point

Mathematics · Calculus & Analysis · ages 20-21

Name ______________________ Date ____________

Nadia says x squared is continuous at 1 because whenever a sequence converging to 1 is squared, the outputs converge to 1. Is her use of the sequential criterion correct?
Circle one: True False
Which function is uniformly continuous on its domain?
- f(x) equal to x squared on all real numbers
- f(x) equal to 2x plus 1 on the real numbers
- f(x) equal to 1 over x on (0, 1)
For f(x) equal to 3x at 2, which delta answers a given epsilon?
- Delta equal to epsilon over 3
- Delta equal to epsilon, which is larger than needed for positive epsilon
- Delta equal to 3, regardless of the requested tolerance
For f(x) equal to 2x at c equal to 1, what is the largest positive delta that works for epsilon equal to 0.2? Give a decimal.
Answer: ______________
Which statement captures uniform continuity on a domain?
- A fresh delta may be chosen separately at every point, even for the same epsilon.
- For each epsilon, one delta works for every pair of domain points.
- The function must be bounded before any delta can be chosen.
- Only polynomial functions can satisfy the definition.
The function f(x) equal to x on [0, 1] attains both a maximum and a minimum.
Circle one: True False
The function f(x) equal to 1 over x on [1, 2] is uniformly continuous. True or false?
Circle one: True False
For f(x) equal to 4x at c equal to 3, which delta works for a given positive epsilon?
- epsilon over 4
- epsilon, which leaves the output error four times too large
- 4, regardless of epsilon
- epsilon squared
A student says x squared must be uniformly continuous on the real numbers because it is continuous at each point. Where is the flaw?
- Pointwise delta may vary with x; uniform delta is shared.
- x squared is discontinuous at zero.
- Uniform continuity requires bounded values, so no unbounded function can qualify.
Which function is continuous but not uniformly continuous on its domain?
- f(x) equal to tan x on (-pi over 2, pi over 2)
- f(x) equal to x on all real numbers, with one shared delta
- f(x) equal to sin x on all real numbers, with bounded slope
- f(x) equal to 5 on all real numbers, with no output change

Answer key

For grown-ups. Fold this page away before handing over the rest.

One delta for every point W1-mt_plWc9raOwz-s1

True · That is exactly the sequential reading of continuity at a point.
f(x) equal to 2x plus 1 on the real numbers · The output gap for 2x plus 1 is twice the input gap, so delta equal to epsilon over 2 works on all real inputs.
Delta equal to epsilon over 3 · The output gap is three times the input gap, so keeping the input within epsilon over 3 keeps the output within epsilon.
0.1 · The output gap is twice the input gap, so the largest working distance is 0.2 over 2, which is 0.1.
For each epsilon, one delta works for every pair of domain points. · For each positive epsilon, one positive delta works for every pair of points in the domain.
True · The closed bounded interval supplies the hypotheses of the extreme value theorem, so both bounds are attained.
True · True. It is continuous on a closed bounded interval, so Heine-Cantor gives uniform continuity.
epsilon over 4 · The output gap is four times the input gap, so delta equal to epsilon over 4 works.
Pointwise delta may vary with x; uniform delta is shared. · Pointwise continuity allows delta to depend on the point, while uniform continuity requires one delta for all real inputs.
f(x) equal to tan x on (-pi over 2, pi over 2) · Tangent is continuous on its open domain, but its slope grows without bound near the missing endpoints, so no shared delta works.