Examples of functions and sequences

[doktorwho]

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.

 

[PeroK]

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.

 

[doktorwho]

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?

 

[PeroK]

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.