2026 MAS241(B) Analysis 1
Final Exam
Problem 3 (25 points). Prove that the function f (x) =
∞
X
n=1
x
is continuous on [0, ∞).
n(x + n)
Solution. Changing the dummy index, we can define the following function gn (x) on [0, ∞) as
gn (x) =
n
X
m=1
1
.
m(x + m)
Since it is a finite summation of continuous functions on [0, ∞), gn (x) is clearly continuous on
[0, ∞) for every n ≥ 1. Obviously, for every positive integer m and every x ∈ [0, ∞), we have
P
1
1
≤ m12 . We have already known that ∞
0 < m(x+m)
m=1 m2 converges. Therefore, gn (x) converges
uniformly on [0, ∞) by the theorem (Theorem 7.10 or what sometimes called Weierstrass M test.) The limit of continuous functions that converges uniformly is continuous (Theorem 7.12)
P
1
so the limit of gn , which is g(x) := ∞
m=1 m(m+x) , is continuous on [0, ∞). Then, f (x) = xg(x)
is the product of two continuous function so that f (x) is continuous on [0, ∞).
—————————————————————————————————————————
P
x
, we need to fix a point and a compact set
If we consider full function hn (x) := nm=1 m(x+m)
containing it in order to use uniform convergence. Fix x0 ∈ [0, ∞) and denote a compact set
C = [0, x0 +1]. Similar reason to above can make a conclusion that hn (x) converges uniformly on
C, and therefore f (x) is continuous on C. By the same theorem above, f (x) is continuous on x0
and we can choose an arbitrary point x0 in [0, ∞), which means that f (x) is continuous on [0, ∞).
Remarks
• Basic criteria : showing uniform convergence (+20 points), and making a conclusion
of continuity of given f (x) by mentioning that f (x) is the uniform convergence limit of
continuous partial sum (+5 points).
• You can prove the continuity directly, not using the uniform continuity. In this case, you
have to show 1. the series given is convergent (so that we can use the linearity of convergent
series), and 2. continuity of f (x) from the definition of continuity and f (x).
• If you considered like as the second solution (using similar function to hn (x)), you have
to use a compact (at least bounded) set, or it would be (−14 points). It is a very-very
important issue so worth taking more than half of full points. The main point is that
hn (x) is NOT uniformly convergent on full x ∈ [0, ∞): if x can be arbitrarily large, then
P
x
the convergence of ∞
m=1 m2 is not uniform, i.e., the δ becomes dependent on not only
ϵ, but also x. Hence, we have to fix some compact set (or at least bounded set), which
confine the range of x in order to show a critical property (uniform convergence.)
• This happens because the uniform convergence is a global property while the continuity
is a local property. So, if you want to show the uniform continuity, you have to check its
convergence regardless the value of x. On the other hand, continuity is only regarding its
open neighborhood so it is okay to see only small open neighborhood. Hence again, the
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2026 MAS241(B) Analysis 1
Final Exam
correct logic is
Partial sum uniformly convergent on every compact (or bounded) set
→ f (x) continuous on that compact (or bounded) set
→ we can choose a compact set containing an arbitrary point we want to see
→ f (x) is continuous on [0, ∞).
• “In the case of the middle of correct proof”, if you used Weierstrass M -test with M
depending on x, it would be (−4 points): in this test, the right-hand side Mn must be
independent on x to show the uniform convergence.
• “In the case of the middle of correct proof”, if you concluded that the given series is
uniformly convergent on [0, ∞) since it is uniformly convergent on [0, b] for every b ≥ 0, it
would be (−7 points).
• Only showing the continuity of f (x) on the bounded set (e.g. only on [0, 1]) would have no
points. Plus, just using M -test also takes no points because this M -test also can be used
to prove its pointwise convergence. You have to connect its result of this test to uniform
convergence with correct logic. Just using M -test is not enough. (Of course, wrong logic
cannot get any points.)
• Each minor misses would get (−2 points).
– You have to mention the continuity of “partial sum,” not just each term because the
uniform convergent of series is a limit of partial sum, not just each term. In this case
it would be considered as minor miss. (Of course, if you did not mention the partial
sum, it would get (−5 points) based on the criteria above.)
– If your answer could be problematic at x = 0, then it would be seen as a minor miss.
– In M −test, we need to use the absolute value or at least mention its non-negativity.
(You have already seen many counterexamples showing that upper bound without
absolute value is not enough to conclude something.) In this case, it would be seen
as a minor miss.
– You have to finalize the proof with arguing that we can choose arbitrary bounded set
so f (x) is continuous on [0, ∞). Or, it would be taken as a minor mistake.
– Simple mistakes (e.g. wrong index/inequality/constant,etc.) would be considered as
minor miss.
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