5.1 Continuity of a Function at a Point 235
16. Prove Corollary 5.1.15, Parts (a) and (b). [Hint: use Example 5.1.9 and
Theorem 5.1.14.]
17. Prove Corollary 5.1.15, Parts (c) and (d). [See Exercise 1.2-B.6.]
18. Suppose that :J K > 0 3 'Vx, y E 'D(J), lf(x) - J(y)J :::; KJx - yJ. Prove
that f is continuous everywhere on its domain.
19. Give an example of a function f : JR--+JR that is discontinuous at every
point of JR, but such that If I is continuous everywhere on JR.
- Use the "algebra of continuous functions" theorem to prove Theorem
5.1.16: tanx, cotx, secx, and cscx are continuous everywhere on their
domains.
- Give examples for each of the following:
(a) functions f,g: JR--+JR, which are discontinuous at every point of JR,
but such that f + g is continuous everywhere on JR.
(b) functions f, g : JR--+JR, which are discontinuous at every point of JR,
but such that f g is continuous everywhere on JR.
- Suppose f : JR--+JR is continuous on JR and a < b. Prove that 1-^1 (a, b)
must be open and 1-^1 [a, b] must be closed, but f(a, b) need not be open.
(In Theorem 5.3.6 we shall prove that f[a, b] must be closed.)
- Suppose f : JR--+JR is continuous on JR. Prove that 'Va E JR,
(a) the sets 1-^1 (-00, a)= {x: f(x) <a} and 1-^1 (a, +oo) are open;
(b) the sets 1-^1 (-00, a]= {x: f(x):::; a} and 1-^1 [a, + oo) are closed;
(c) the set 1-^1 (a) = {x: f(x) =a} is closed.
- Suppose f , g : JR--+JR are continuous on JR. Prove that
(a) the set {x: f(x) = g(x)} is closed;
(b) the set {x: f(x):::; g(x)} is closed;
(c) the set {x: f(x ) < g(x)} is open.
- Local Boundedness Property: Prove that if f is continuous at xo then
f is locally bounded at xo (i.e., bounded on some neighborhood of xa).
That is, :le> 0 and :JM> 0 3 'Vx E 'D(J) n Nro(xo), lf(x)I:::; M.
- Neighborhood Inequality Property of Continuous Functions:
Prove that if f is continuous and positive at xo, then f is positive in
some neighborhood of x 0. In fact, if f is continuous at Xo and f (xo) > c,
then :J neighborhood N8(xo) 3 'Vx E 'D(J) n N8(xo), f(x) > c. State and
prove an analogous result for f continuous at Xo and f (xo) < c.
