Calculus Volume 1 · 免费预览
连续性
函数在其图像连续无间断之处是连续的，在代数上则由极限与函数值相等来定义，而连续性定理保证了初等函数及其复合函数所具有的这些性质。
1. The definition of continuity
The definition of continuity
2. Continuity theorems
Continuity theorems
3. The intermediate value theorem
The intermediate value theorem
4. Continuity and the limit laws together
Continuity and the limit laws together
5. Three conditions for continuity
Which trio of conditions together defines continuity of f at the point a?
6. Classifying a hole
The function g equals (x^2 - 1)/(x - 1) for x not 1 and is left undefined at x = 1. What kind of discontinuity does g have at x = 1, and can it be repaired?
7. Intermediate value theorem hypothesis
The intermediate value theorem promises a solution of f(c) = N between a and b. Which hypothesis is essential?
8. Pulling a continuous function through a limit
Which computation correctly evaluates lim x to 0 of e^(sin x)?
9. Continuity at a piecewise join
The function f(x) = x^2 for x at most 1 and f(x) = 2x - 1 for x greater than 1 is tested at the join x = 1. Which statement is correct?
10. Extreme value theorem
A function is continuous on the closed interval [0, 3]. Which guarantee does the extreme value theorem provide?
11. Continuity of polynomials
Every polynomial is continuous on the whole real line. How many discontinuities does p(x) = x^2 + 1 have? Answer with a number.
12. Intermediate value target
f(x) = x^3 - x - 1 satisfies f(1) = -1 and f(2) = 5. By the intermediate value theorem, f attains the value 0 somewhere in (1, 2). What value does the theorem guarantee is attained? Answer with a number.
13. Removable value
The function (x^2 - 1)/(x - 1) has limit 2 at x = 1. Defining g(1) equal to that limit removes the discontinuity. What value should g(1) be? Answer with a number.
14. Jump size
A step function has left limit 2 and right limit 5 at x = 1. What is the size of the jump, the right limit minus the left limit? Answer with a number.