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Real analysis

  • #1

Homework Statement


Mod note: Edited the function definition below to reflect the OP's intent.
Suppose f:R->R is continuous. Let λ be a positive real number, and assume that for every x in R and a>0,f(ax)=aλ f(x). (a) If λ > 1 show that f is differentiable at 0. (b) If 0 < λ < 1 show that f is not differentiable at 0. (c) If λ=1, show that f is differentiable at 0 if and only if it is linear. (Hint: what is f(0)?)

Homework Equations




The Attempt at a Solution


I am considering start the question with f(x)-f(0)/x but how can i find the limit when x approaches 0? f(x)=f(1*x)=1^λf(x) f(0)=f(0*x)=0^λf(x)=0, then is f(x)-f(0)/x = f(x)/x? Then how can i know it's differentiable at 0?
 
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Answers and Replies

  • #2
15
3
You must set up a limit, epsilon delta style
 
  • #3
You must set up a limit, epsilon delta style
how can i use epsilon delta to prove differentiability?
 
  • #4
33,174
4,858

Homework Statement



Suppose f:R->R is continuous. Let λ be a positive real number, and assume that for every x in R and a>0,f(ax)=aλ f(x). (a) If λ > 1 show that f is differentiable at 0. (b) If 0 < λ < 1 show that f is not differentiable at 0. (c) If λ=1, show that f is differentiable at 0 if and only if it is linear. (Hint: what is f(0)?)

Homework Equations




The Attempt at a Solution


I am considering start the question with f(x)-f(0)/x
You are missing parentheses. What you wrote is ##f(x) - \frac{f(0)}{x}##. I'm pretty sure you meant ##\frac{f(x) - f(0)}{x}##. If you don't use LaTeX like I did, you need to write it as (f(x) - f(0))/x.
silvetriver said:
but how can i find the limit when x approaches 0? f(x)=f(1*x)=1^λf(x) f(0)=f(0*x)=0^λf(x)=0, then is f(x)-f(0)/x = f(x)/x? Then how can i know it's differentiable at 0?
What does ^ mean here? It is often used to indicate exponentiation, as in 2^3 = 8.
Are you using it to indicate "and"?
 
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  • #5
You are missing parentheses. What you wrote is ##f(x) - \frac{f(0)}{x}##. I'm pretty sure you meant ##\frac{f(x) - f(0)}{x}##. If you don't use LaTeX like I did, you need to write it as (f(x) - f(0))/x.
What does ^ mean here? It is often used to indicate exponentiation, as in 2^3 = 8.
Are you using it to indicate "and"?
You are right thats what i mean. yea ^ means exponentiation...
 
  • #6
15
3
"how can i use epsilon delta to prove differentiability?"

Use the definition of the derivative.
 
  • #7
33,174
4,858
You are right thats what i mean. yea ^ means exponentiation...
Then I'm really confused now. Where does this come from?
f(x)=f(1*x)=1^λf(x) f(0)=f(0*x)=0^λf(x)=0
I don't see anything in the problem statement about exponents.

In any case, the above is very ambiguous. For example you have 1^λf(x), which I take to mean ##1^{\lambda}f(x)##, according to the normal rules for the order of operations. I don't know why you would have ##1^{\lambda}##, which equals 1, or ##0^{\lambda}##, which equals 0, as you have elsewhere.
 
  • #8
Then I'm really confused now. Where does this come from?

I don't see anything in the problem statement about exponents.

In any case, the above is very ambiguous. For example you have 1^λf(x), which I take to mean ##1^{\lambda}f(x)##, according to the normal rules for the order of operations. I don't know why you would have ##1^{\lambda}##, which equals 1, or ##0^{\lambda}##, which equals 0, as you have elsewhere.
oh sorry i mistyped the problem. it should be f(ax)=a^lambda f(x) I think 1^lambda=1 because the problem states that lambda>1
 
  • #9
15
3
eq0002MP.gif
' of course the a in this definition is not the constant in the definition of f(x)
 
  • #10
33,174
4,858
oh sorry i mistyped the problem. it should be f(ax)=a^lambda f(x) I think 1^lambda=1 because the problem states that lambda>1
Is it safe to interpret the above as ##f(ax) = a^{\lambda}f(x)##?
 
  • #11
Is it safe to interpret the above as ##f(ax) = a^{\lambda}f(x)##?
yessss i just realize that f(0)=f(a*0)=a^lambda f(0), well then i totally get lost...
 
  • #12
eq0002MP.gif
' of course the a in this definition is not the constant in the definition of f(x)
yea thats what i think to start with but i can't find f(x) and f(0)
 
  • #13
33,174
4,858
You must set up a limit, epsilon delta style
The OP has to set up a limit, but I don't believe that a delta-epsilon argument is necessary.
 
  • #14
15
3
what i think
The nice thing about the definition of derivative AND f(x) is that you only need to find the difference, not the actual function.
 

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