Convergence of a continuous function related to a monotonic sequence

In summary, the homework statement is that if a function is continuous then for every monotonic sequence in the domain that converges to a given point, the limit exists. The theorem 11.4 states that every sequence has a monotonic subsequence. To prove the theorem, the author uses the axiom of choice and constructs a sequence that does not converge to the original sequence, but instead converges to the limit.
  • #1
fishturtle1
394
82

Homework Statement


Let ##f## be a real-valued function with ##\operatorname{dom}(f) \subset \mathbb{R}##. Prove ##f## is continuous at ##x_0## if and only if, for every monotonic sequence ##(x_n)## in ##\operatorname{dom}(f)## converging to ##x_0##, we have ##\lim f(x_n) = f(x_0)##. Hint: Don't forget Theorem 11.4.

Homework Equations


Theorem 11.4: Every sequence has a monotonic subsequence.

Theorem 11.8: ##(s_n)## has a sub sequential limit ##L \epsilon \mathbb{R}## and ##(s_n)## converges, then ##\lim s_n = L##.

The Attempt at a Solution


Proof: ##\rightarrow## Suppose ##f## is continuous. Then for all sequences ##(x_n)## in ##\operatorname{dom}(f)## that converge to ##x_0##, we have that ##\lim f(x_n) = f(x_0)##. A monotonic sequence ##(a_n)## in ##\operatorname{dom}(f)## that converges to ##x_0## is a sequence in ##\operatorname{dom}(f)## that converges to ##x_0##. So for all monotonic sequences ##(a_n)## in ##\operatorname{dom}(f)## that converge to ##x_0##, we have ##\lim f(a_n) = f(x_0)## as desired.

##\leftarrow## Suppose for all monotonic sequences ##(a_n)## in ##\operatorname{dom}(f)## that converge to ##x_0##, we have ##\lim f(a_n) = f(x_0)##. Suppose ##(a_m)## is a sequence in ##\operatorname{dom}(f)## such that ##\lim a_m = x_0##. By 11.4 and 11.8, we have ##a_m## has a monotonic subsequence ##(a_{m_k})## that converges to ##x_0## and ##\lim f(a_{m_k}) = f(x_0)##. We also have the sequence ##(b_k)## defined by ##b_k = f(a_{m_k})## is a subsequence of the sequence ##(b_m)## defined by ##b_m = f(a_m)##. So ##(f(a_m))## has a sub sequential limit ##(f(x_0))##. We need to show ##(f(a_m))## converges...

Also I'm not sure if when I want to say a function converges since a function is a set, Do i have to define a new sequence like ##(b_k)## defined as ##b_k = f(x)##, or Should I write it as ##(f(x))##?
 
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  • #2
For the second direction, picking a sequence is not going to help, as showing it converges will not lead anywhere.
Instead, assume ##f## is not continuous and use that to construct a sequence ##(x_n)## that converges to ##x_0## such that ##f(x_n)## does not converge to ##f(x_0)##. That will prove what we want by contradiction.

It is easy to do using the axiom of choice. Start by observing that if ##f## is not continuous at ##x_0## then there exists some ##\epsilon>0## such that for any ##\delta##, there exists ##x\in(x_0-\delta,x_0+\delta)## such that ##|f(x)-f(x_0)|>\epsilon##. Think of a way to get a sequence of progressively smaller deltas.

Doing it without the axiom of choice may be more difficult (I'm not even sure if it's possible). But the question doesn't say the axiom may not be used.
 
  • #3
andrewkirk said:
For the second direction, picking a sequence is not going to help, as showing it converges will not lead anywhere.
Instead, assume ##f## is not continuous and use that to construct a sequence ##(x_n)## that converges to ##x_0## such that ##f(x_n)## does not converge to ##f(x_0)##. That will prove what we want by contradiction.

It is easy to do using the axiom of choice. Start by observing that if ##f## is not continuous at ##x_0## then there exists some ##\epsilon>0## such that for any ##\delta##, there exists ##x\in(x_0-\delta,x_0+\delta)## such that ##|f(x)-f(x_0)|>\epsilon##. Think of a way to get a sequence of progressively smaller deltas.

Doing it without the axiom of choice may be more difficult (I'm not even sure if it's possible). But the question doesn't say the axiom may not be used.
i'm not sure how to use the axiom of choice here. The wikipedia entry says for any set X of nonempty sets, there exists a choice function f defined on X. But that doesn't seem applicable because we don't have a set of nonempty sets?

Going off post #2:
if ##f## is not continuous at ##x_0## then there exists some ##\epsilon>0## such that for any ##\delta##, there exists ##x\in(x_0-\delta,x_0+\delta)## such that ##|f(x)-f(x_0)|>\epsilon##. By 11.4, there is some monotonic subsequence ##(x_{n_k})## of ##(x_n)##. By 11.8, ##(x_{n_k})## converges to ##x_0##. I think that ##x_{n_k}## being monotonic gives us smaller deltas because we are getting closer and closer to the limit but I'm not sure how to put this in words.
 
  • #4
For simplicity of notation, use the standard 'ball' notation that ##B_r(x)## denotes the 'open ball' of radius ##r## centred at ##x## which, for ##\mathbb R## means that ##B_\delta(x)## is the interval ##(x-\delta,x+\delta)##. Now the sequence ##(\delta_n)## such that ##\delta_n=1/n## is monotonically reducing to 0.

Also, the collection

$$C = \{B_{\delta_n}(x_0)\ :\ \delta_n=1/n\}$$

is a collection of nonempty sets. Define ##A_n## to be the set of points ##x'## in ##B_{\delta_n}(x_0)## such that ##|f(x')-f(x_0)|>\epsilon##. Do we know whether any given set ##A_n## is empty or not, given what we assumed about ##f## not being continuous? If we were able to conclude that every ##A_n## is nonempty, can we say anything about whether there exists a sequence ##(x_n)## such that ##x_n\in A_n## (think axioms)? To what would that sequence converge? Would ##(f(x_n))## converge to ##f(x)##? Can we get a contradiction?
 

Related to Convergence of a continuous function related to a monotonic sequence

1. What is the definition of convergence for a continuous function related to a monotonic sequence?

Convergence of a continuous function related to a monotonic sequence is a mathematical concept that describes the behavior of a sequence of real numbers that approaches a certain limit. In this case, the sequence is monotonic, meaning that it either always increases or always decreases. The function is continuous, meaning that it has no abrupt changes or discontinuities. In simpler terms, convergence means that the values of the function get closer and closer to a specific value as the sequence progresses.

2. How is convergence of a continuous function related to a monotonic sequence different from general convergence?

Convergence of a continuous function related to a monotonic sequence is a specific type of convergence that applies to a certain type of sequence and function. General convergence, on the other hand, applies to any type of sequence and function. The key difference is that in the case of convergence related to a monotonic sequence and continuous function, the sequence must be monotonic and the function must be continuous for convergence to occur.

3. What are some examples of a monotonic sequence and a continuous function?

An example of a monotonic sequence is {1, 2, 3, 4, ...} which always increases, or {10, 8, 6, 4, ...} which always decreases. An example of a continuous function is f(x) = x^2, which has no abrupt changes or discontinuities and can be graphed as a smooth curve.

4. How can you determine if a continuous function related to a monotonic sequence is convergent?

To determine if a continuous function related to a monotonic sequence is convergent, you can use the Monotone Convergence Theorem. This theorem states that if a sequence is monotonic and bounded (meaning it has an upper and lower bound), then it must converge to a limit. Additionally, you can also use the definition of convergence to check if the values of the function are getting closer and closer to a specific value as the sequence progresses.

5. What are some real-life applications of convergence of a continuous function related to a monotonic sequence?

Convergence of a continuous function related to a monotonic sequence has many practical applications in various fields, such as physics, economics, and engineering. For example, it can be used to model population growth, predict stock market trends, and analyze the stability of physical systems. In everyday life, it can also help us understand and make predictions about natural phenomena, such as the movement of tides or the growth of plants.

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