Worksheet: Week 4
Foundations
Definition (Limit, ε–δ)
Let f be defined on an open interval containing a (except possibly at a). We say
if and only if for every ε > 0 there exists a δ > 0 such that
0 < |x − a| < δ ⇒ |f(x) − L| < ε.
Definition (Continuity at a point)
A function f is continuous at a if
A function is continuous on an interval if it is continuous at every point of that interval. Let’s use the symbol ∀ to denote the phrase for all and ∃ to denote the phrase there exists. We can formulate the formal definition of continuous function on [a,b]:
f is continuous on the interval [a,b] ⇐⇒
∀c ∈ [a, b], ∀ε > 0, ∃δ > 0 such that ∀x ∈ [a, b], |x − c| < δ ⇒ |f(x) − f(c)| < ε
Toy example: show f(x) = x is continuous on [0, 1] (via ε–δ)
Fix a ∈ [0, 1]. We must show: for every ε > 0 there exists δ > 0 such that |x − a| < δ implies |f(x) − f(a)| < ε.
Here f(x) = x, so |f(x) − f(a)| = |x − a|. Choose δ = ε. Then whenever |x − a| < δ = ε we have
|f(x) − f(a)| = |x − a| < ε,
as required. Since a was arbitrary, f(x) = x is continuous on [0, 1].
Practice
1. Describe what is the meaning of x3
is continuous on the interval [0, 2] using the formal definition (i.e. Do not include the word limit or lim).
2. Let a and x be such that a ∈ [0, 2], x ∈ [0, 2]; Find the maximum value of x2 + ax + a2
(Ans: it is 12).
3. Use the factorization |(x3 − a3)| = |x − a| × |x2 + ax + a2| and part (2) to show that x3
is continuous on the interval [0, 2] (Notice that the quantity x2 + ax + a2
is positive, so you can remove the absolute value)
Remark: In general xn − an = (x − a)(xn−1 + axn−2 + a2xn−3 + ... + an−1). The simplest case is n = 2 → x2 − a2 = (x − a)(x + a) and n = 3 as in part (3).
True/False (T F Circle one and justify briefly.)
1. If f and g are continuous on [a, b], then
T F
2. If f and g are continuous on [a, b], then
T F
3. If f is continuous on [a, b], then
T F
4. If f is continuous on [a, b], then
T F
5. If f is continuous on [a, b] and f(x) ≥ 0, then
T F
6.
T F
7. If f′
is continuous on [1, 3], then
T F
8. If v(t) is the velocity at time t of a particle moving along a line, then
is the distance traveled during a ≤ t ≤ b.
T F
9.
T F
10. If f and g are differentiable and f(x) ≥ g(x) for a < x < b, then f′
(x) ≥ g′
(x) for a < x < b.
T F
11. If f and g are continuous and f(x) ≥ g(x) for a ≤ x ≤ b, then
T F
12.
T F
13. All continuous functions have derivatives.
T F
14. All continuous functions have antiderivatives.
T F
15.
T F
16. If
dx = 0, then f(x) = 0 for 0 ≤ x ≤ 1.
T F
17. If f is continuous on [a, b], then
T F
18.
dx represents the area under the curve y = x − x3
from 0 to 2.
T F
19.
T F
20. If f has a discontinuity at 0, then
f(x) dx does not exist.
T F