Practice question · Select all that apply
f is continuous on the closed interval from 0 to 4. Select every statement that MUST be true.
Hints
- Two cornerstone theorems apply on a closed bounded interval; a third property is strictly stronger than continuity.
- A root requires the values to straddle zero, which was never assumed.
Show the answer
- A. f attains a minimum value on that interval
- B. f takes every value between f(0) and f(4)
- C. f attains a maximum value on that interval
Why
The Extreme Value Theorem supplies a maximum and a minimum, and the Intermediate Value Theorem supplies every intermediate value. But a root needs a SIGN CHANGE, which was not assumed, the constant function 7 has none, and continuity never implies differentiability, as the absolute value shows.
Practise Continuity
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More questions on Continuity
- Continuity at a point requires three separate conditions, and the limit existing is only one of them.
- Sort each function by whether it is continuous at every real number.
- Order the checks that establish continuity of f at the point a.
- The function |x| is continuous everywhere and has no derivative at 0. What does that pair of facts establish…