Calculus: Analytic Geometry and Calculus, with Vectors

3.4 Continuity 151

As is easy to guess, the vector function r is said to be continuous at t if

(5)

and we write

(6)

if w is a vector for which

(7)

Limo Ir(t + At) - r(t) I = 0,

lim r(t) = w

t--.to

lim Ir(t) - wI = 0.

t-.to

It is a consequence of (4) that a vector function is continuous if and only if its
scalar components are continuous.
12 Using ideas from the preceding problem, let
r(t) = x(t)i + y(t)j + z(t)k
w=ai+bj+ck

and prove that lim r(t) = w if and only if

t-to

lim x(t) = a, lim y(t) = b, lim z(t) = c.

t-+to t-+to t--.to

Hint: Write and use a formula for Ir(t) - wI.

13 Once again, let the symbol [q] denote the greatest integer which is less

than or equal to q. Let f be the function for which

when x > 0. Draw the graph off and tell where f is discontinuous.

14 Using the "bracket notation" of the preceding problem, determine whether

lim xr
z-.o+ Lx1J
exists.
T5 Letting D be our old friend, the dizzy dancer function, for which
D(x) = 0 (x irrational)
D(x) = 1 (x rational),

show that there is no a for which

lim D(x) = D(a)

a-.a

and hence that this function is everywhere discontinuous.
16 A potential new friend g is defined over the closed interval 0 5 x 5 1 in
an interesting way. If x is irrational, then g(x) = 0. If x is 0, then g(x) = 1,
and if x = 1, then g(x) = 1. If x is a rational number for which 0 < x < 1 and
if x = m/n, where m and n are positive integers having no common positive
integer factor exceeding 1, then g(x) = 1/n. Thus g(y) = $,g(() = e,g(*) = 3,
g(a) = +, g(') = ,g(3) _ 1, g(3) = 3, etcetera. Sketch a figure indicating the