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Uniform convergence and differentiation
32 Investigate the convergence of the sequence of functions defined by
f
( x ) x /(1 nx )onR.
1 nx
(1 nx )
Provethat f
( x )
.
If f is thepointwiselimit function, show that
sup
f
( x )
f ( x )
x R
1/2
n , and deduce that the convergence
is uniform.
Deduce that lim
f
(0)
f
(0).
Is lim
f
f ? Useqn 24.
Question 32 shows that differentiability is less well behaved under
uniform convergence than is integrability.
33 Let f
( x ) (cos nx )/ n .
Find the pointwise limit function for the sequence ( f
) and
determine whether the convergence is uniform on R.
Determine the function f
( x ). Is there a pointwise limit function for
the sequence ( f
)? Consider values of x 0. Determine the value of
f
. Is lim
f
lim
f
?
From qns 32 and 33 it is clear that, even when the sequence ( f
)
converges uniformly to f and every function f
is differentiable, the
sequence ( f
) need not be convergent and when it is convergent it may
not be uniformly convergent. If we are to formulate a theorem about
the uniform convergence of a sequence of derivatives, the uniform
convergence of the sequence of derivatives will have to be assumed.
Then, if the functions under discussion are all continuous, the
Fundamental Theorem of Calculus may enable us to use the good
behaviour of integrals under uniform convergence to claim results
about derivatives.
34 We suppose that we have a sequence of differentiable functions ( f
)
on an interval [ a , b ] converging to the pointwise limit function f .
Wefurthr suppose
(a) that the sequence ( f
) consists of continuous functions, and
(b) that the sequence ( f
)
uniformly.
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