For Wednesday: do true/false questions for chapter 4.
Also, does there exist a twicedifferentiable function
f
on
R
satisfying
f
(
x
) < 0 and
f
′′
(
x
) > 0 for all
x
?
Section 4.7: Antiderivatives (concluded)
Last time we asked “Does the function
f
(
x
) = 
x
 have an
antiderivative on
R
?” and found that the answer was Yes;
the general form was …
..?.
.
F
(
x
) =
x

x
 / 2 +
C
.
Does the Heaviside function
{0 if
x
< 0
f
(
x
)
=
{
{1 if
x
≥
0
have an antiderivative on
R
? .
..
..?.
.
No.
Why not?
..?.
.
Is it enough to say “The Heaviside function is
discontinuous, so it doesn’t have an antiderivative on the
whole real line
R
?” .
..
No, because some discontinuous functions have
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View Full Documentantiderivatives (see the discussion of the badly behaved
function 2
x
sin (1/
x
) – cos (1/
x
) earlier); that is, a
discontinuous function can still be the derivative of some
differentiable function.
Suppose there were an antiderivative of
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 Fall '11
 Staff
 Calculus, Antiderivatives, Derivative, Continuous function, Heaviside function

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