Examples of functions and sequences

In summary: If you want a challenge, how about trying to find a function that is continuous on all of ##\mathbb{R}## but is discontinuous at almost every point -- i.e. it's only continuous on a set of measure zero?
  • #1
doktorwho
181
6

Homework Statement


Give the example and show your understanding:
[1][/B].Lets define some property of a point of the function:
1. Point is a stationary point
2. Point is a max/min of a function
3. Point is a turning point of a function
If possible name a function whose point has properties of being:
a) 1&2
b) 2&3
c) 1&3
d) 1&2&3
[2]. Give examples of a sequence ##{c_n}## that has these properties:
a) The limit of a sequence is +∞
b) The sequence has limit of 0 but can be separated into 2 sequence products of which one diverges
c) The sequence is divergent with at least one of its limits going to +∞
[3]. Name a function and an interval on which it is (if possible):
a) Continuous and unbounded
b) Continuous and Bounded
c) Discontinuous and Unbounded
d) Discontinous and Bounded
e) Unbounded and differentiable
f) Bounded and differentiable
g) Unbounded and undifferentiable
i) Bounded and undifferentiable

Homework Equations


3. The Attempt at a Solution [/B]
Here are my examples for all 3 parts and i would really appreciate your feedback on each.
[1].
a) ##f(x)=x^2##
b) [could not find such function, is this even possible?]
c) ##f(x)=x^3##
d) [same as with the b), i could not find one. I tried ##cosx## but its turning point is not its max]

[2].
a) ##c_n=n##
b) ##c_n=\frac{\sin x}{x}##
c)##c_n=(-2)^n##

[3]
a) ##f(x)=x## (R)
b) ##f(x)=\arcsin x## [-1,1]
c) ##f(x)= 1/x## not sure about the interval, maybe [-1,1]
d) could not find
e) ##f(x)=x## on R
f) ##f(x)=1/x## on [-1,1]
g) could not find
i) could not find
Appreciate it, thanks.
 
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  • #2
doktorwho said:

Homework Statement


Give the example and show your understanding:
[3]. [/B]Name a function and an interval on which it is (if possible):
a) Continuous and unbounded
b) Continuous and Bounded
c) Discontinuous and Unbounded
d) Discontinous and Bounded
e) Unbounded and differentiable
f) Bounded and differentiable
g) Unbounded and undifferentiable
i) Bounded and undifferentiable

[3]
a) ##f(x)=x## (R)
b) ##f(x)=\arcsin x## [-1,1]
c) ##f(x)= 1/x## not sure about the interval, maybe [-1,1]
d) could not find
e) ##f(x)=x## on R
f) ##f(x)=1/x## on [-1,1]
g) could not find
i) could not find
Appreciate it, thanks.

There's far too much here for one post. I've just focused on 3:

c) is not correct. The function ##1/x## is continuous (on the domain on which it is defined), but it's not defined on all of ##[-1, 1]##.

d) should be easy. All you need is a discontinuity. Try defining your function piecewise.

f) ##1/x## is not defined on all of ##[-1, 1]##

g/i) It shouldn't be hard to find a function that is not differentiable. Remember that a function doesn't have to have a "nice" formula.
 
  • #3
PeroK said:
There's far too much here for one post. I've just focused on 3:

c) is not correct. The function ##1/x## is continuous (on the domain on which it is defined), but it's not defined on all of ##[-1, 1]##.

d) should be easy. All you need is a discontinuity. Try defining your function piecewise.

f) ##1/x## is not defined on all of ##[-1, 1]##

g/i) It shouldn't be hard to find a function that is not differentiable. Remember that a function doesn't have to have a "nice" formula.

Well then i could use for f) ##arctanx## on (R) and for g could i use g) ##arctan|x|## on the same interval?
and for c) is ##sgnx## on R ok?
 
  • #4
doktorwho said:
Well then i could use for f) ##arctanx## on (R) and for g could i use g) ##arctan|x|## on the same interval?
and for c) is ##sgnx## on R ok?

Functions don't have to have a single formula. For example, the following function is discontinuous (it's called a step function):

##f(x) = 0 \ (x \le 0) \ ; f(x) = 1 \ (x > 0)##

This is called defining a function piecewise. Some people call these "piecewise" functions, but on a pedantic note there is no such thing. It's defined piecewise, but it's a regular function.

You can make functions more exotic. For example:

##f(x) = 0 \ (x \in \mathbb{Q}) \ ; f(x) = 1 \ (x \notin \mathbb{Q})##

What can you say about the continuity of that function?

Another trick is to fill the gap where a function is not defined. E.g.:

##f(x) = 0 \ (x = 0) \ ; f(x) = 1/x \ (x \ne 0)##

Can you use that function in any of your cases?

I did think that your exerise might require finite intervals (so ##\mathbb{R}## would not be allowed). Perhaps you should check.

In any case, you need to move on from needing a single formula for each function and add the ones I've posted to your bag of tricks.
 

Related to Examples of functions and sequences

1. What is a function?

A function is a mathematical rule or relationship that assigns each input value to exactly one output value. It can be represented by an equation, table, or graph.

2. How is a function different from a sequence?

A function is a rule that maps a specific set of input values to output values, while a sequence is an ordered list of numbers or objects. A function can be thought of as a special type of sequence where each term is related to the previous one by a specific rule.

3. What are some real-world examples of functions and sequences?

Functions can be used to model various real-world phenomena such as population growth, temperature changes, and financial investments. Sequences can be found in everyday situations like counting numbers, musical notes, and sports rankings.

4. How do you determine the domain and range of a function?

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. To determine the domain, look at the restrictions on the input values, such as domain restrictions in a rational function or the input values listed in a table. To determine the range, consider the possible output values based on the function's equation or graph.

5. Can a sequence be infinite?

Yes, a sequence can be infinite if it continues indefinitely without repeating or approaching a specific value. An example of an infinite sequence is the Fibonacci sequence, where each term is the sum of the two previous terms.

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