Chapter 14

Vector Valued Functions Of OneVariable

14.1 Limits Of A Vector Valued Function Of One RealVariable

As in the case of a scalar valued function of one variable, the derivative is defined as

limh→0

f (t0 +h)−f (t0)h

.

Thus the derivative of a function of one variable involves a limit. The following is thedefinition of what is meant by a limit. The new topic is the case of one sided limits althoughthere is really nothing essentially new from what was done earlier. Here is the definition.

Definition 14.1.1 In the case where D(f) is only assumed to satisfy D(f)⊇ (t, t + r),

lims→t+

f (s) =L

if and only if for all ε > 0 there exists δ > 0 such that if

0 < s− t < δ ,

then|f (s)−L|< ε.

In the case where D(f) is only assumed to satisfy D(f)⊇ (t− r, t),

lims→t−

f (s) =L

if and only if for all ε > 0 there exists δ > 0 such that if

0 < t− s < δ ,

then|f (s)−L|< ε.

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