Page 212 - DMTH401_REAL ANALYSIS

Real Analysis

Notes The sequence of derivatives is
f‘(x) = cos(nx)
Which does not converge (take for example x = ).

Example: Find a sequence of differentiable functions that converges uniformly to a
continuous limit function but the limit function is not differentiable

we found a sequence of differentiable functions that converged point wise to the continuous,
non-differentiable function f(x) = |x|. Recall:

ì 1 1
x
1
ï - x - 2n if -   - n
ï
ï n 2 1 1
f (x)= í x if - < x <
n 2 n n
ï
ï 1 1
ï x - if  x  1
î 2n n
That same sequence also converges uniformly, which we will see by looking at ‘|| f – f|| .
n D
We will find the sup in three steps:
If 1  x –1/n:
f (x) – f(x)| = |–x – / + x| = /
1
1
n 2n 2n
If –1/n < x < 1/n:
1
2
3
n
2
|f (x) – f(x)| | / x | + |x|  / / + / = /
n
1
n 2 2 n n 2n
If 1/n x  1:
1
|f (x) – f(x)| = |x – / – x| = /
1
n 2n 2n
Thus, || f – f|| < / which implies that f converges uniformly to f. Note that all f are
3
n D 2n n n
continuous so that the limit function must also be continuous (which it is). But clearly f(x) = |x|
is not differentiable at x = 0.
16.3 Uniform Convergence and Integrability
¥
Theorem 2: Let { }f n n= 1 be a sequence of real-valued functions on [ , ].a b If all f are Riemann-
n
integrable and f ® f uniformly then f is Riemann-integrable and
n
b b
a ò f n ( )dx ® a ò f ( ) .
dx
x
x
n
:
Proof: Recall partition P a t= 0 < t < ... t = . b The upper Darboux sum ( , )U f p = å i= 1 sup [ i t - 1 , i t ]
<
n
1
x
)
f ( )(t - t i- 1 and the lower Darboux sum ( , )L f P = å n i= 1 inf [ i t - 1 , i t ] f ( ) (t - t i- 1 ). f is Riemann-
x
i
i
integrable on [ , ]a b if and only if " > 0 PU ( , ) L ( , ) <  .
f
f
-
P
$
P
Claim 1: f is Riemann-integrable.

Let  > 0. Since f ® f uniformly n$ s.t. f - f sup < 4(b a ) . Since f is R-integrable P$ s.t.
n
n
n
-
f
P
U ( , ) L ( , ) <  /2. We have
f
P
-
n
n
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