234 Chapter 5 • Continuous Functions
- Sketch the graph of j(x) = Jx3 + 2x^2 + x on the interval [-2, 2]. Is
f continuous at -1? Is lim f(x) = f(-1)? Justify your answers, and
X->-1
reconcile them with the claims made in Exercises 1 and 2. - Use Definition 5.1.l to prove that the function J(x) = 2x^2 + 3x + 5 is
continuous at xo = -1. - Use Definition 5.1.l to prove that the function f(x) = 4x^2 - 5x - 3 is
continuous at Xo = 2. - Prove Theorem 5.1.3.
- Prove Theorem 5.1.7.
- Prove Theorem 5.1.8.
- Prove the claims made in Example 5.1.9 (a). [See Theorems 4.2.l and
4.2.11.] - Prove that the signum function defined in Example 5.1.5 is continuous
everywhere except at x = 0.
ll. Let f(x) = {sin(~) ~f x # 0} and g(x) = { xsin (~) ~f x ;;6 O }·
0 tlx=O 0 tlx=O
Which one or more of these functions is (or are) continuous at x = O?
Justify your answer. [See Examples 4.1.12 and 4.2.21.]12. Consider. the funct10n. f(x) = { x. if x. is. rational. }. Prove that f is.
-x if x is irrat10nal
continuous only at 0. [See Exercise 4.2.18.]- Use Theorems 5.1.13- 5.1.16 to determine the intervals over which the
following functions are continuous:
(a) f(x) = vx^2 + 5
x-1
(c) f(x) = x 2 +2x-3
( e) f ( x) = 3x + 5
x^2 - 4
(g) f(x) = cosy'x
(i) f(x) = tanx(b) f(x) = v3x - 2
x
(d) f(x) = x2+1( f) f ( x) = sin ( ~ ~ ~ )
(h) f(x) = Jcosx
(j) J(x) =tan (sinx)- Prove Theorem 5.1.13. [Hint: See how Theorem 4.2.11 was proved, or use
the sequential criterion.] - Prove Theorem 5.1.14 (c).