Prove That if F is Continuous on r and F X Y F X F Y for Then F X Ax for Some
Examples which satistfy f(x+y)=f(x) + f(y)
- Thread starter HappyN
- Start date
Give two different examples of f:R->R such that f is continuous and satisfies f(x+y)=f(x)+f(y) for every x,y e R. Find all continuous functions f:R->R having this property. Justify your answer with a proof.
I came up with one example:
f(x)=ax
then f(x+y)=a(x+y)=ax+ay=f(x)+f(y)
however, I cant seem to think of another example, any hints?
Answers and Replies
To prove that is straightforward but tedius. Here's an outline of how I would do it.
1) Prove, by induction, that f(nx)= nf(x) for all real numbers, x, and n any positive integer.
2) Use f((0+ n)x)= f(0x)+ f(nx) to show that f(0x)= 0= 0f(x).
3) Use f((n+(-n))x)= f(0)= 0 to show that f(-nx)= -f(nx) and so -nf(x) for any positive integer n.
3) Use f(nx)= nf(x) to show that f(n(1/n)y)= nf((1/n)y)= f(y) so f((1/n)y)= (1/n)f(y).
4) Use f(nx)= nf(x) to show that f((m/n)y)= mf((1/n)y)= (m/n)f(y) for any rational number m/n and any real number y.
5) Use continuity to show that f(rx)= rf(x) for any real numbers r and x and, taking x= 1, that f(r)= rf(1)= ar where a= f(1). (If r is any real number, there exist a sequence of rational numbers, [itex]\{r_n\}[/itex], converging to r. By continuity, [itex]f(rx)= f((\lim r_n)x)= \lim f(r_nx)= (\lim r_n) f(x)[/itex].)
That means, of course, that the graph of a continuous function satisfying f(x+y)= f(x)+ f(y) is a straight line through the origin. There do, however, exist non-continuous functions satisfying that equation- and the graph of such a function is dense in the plane!
That means, of course, that the graph of a continuous function satisfying f(x+y)= f(x)+ f(y) is a straight line through the origin. There do, however, exist non-continuous functions satisfying that equation- and the graph of such a function is dense in the plane!
Only if you accept the axiom of choice!
The question states:
Give two different examples of f:R->R such that f is continuous and satisfies f(x+y)=f(x)+f(y) for every x,y e R. Find all continuous functions f:R->R having this property. Justify your answer with a proof.I came up with one example:
f(x)=ax
then f(x+y)=a(x+y)=ax+ay=f(x)+f(y)however, I cant seem to think of another example, any hints?
try doing it by taking a Newton quotient.
f(x+h) -f(x)/h = f(h)/h so the Newton quotient is constant for all x.
So there is a derivative and it must be constant. Since f(0) = 2f(0) the function must be ax for some a.
HOW do you assert "so there is a derivative"? There is a derivative if and only if [itex]\lim_{h\to 0} f(h)/h[/itex] exists but how do you show that?try doing it by taking a Newton quotient.f(x+h) -f(x)/h = f(h)/h so the Newton quotient is constant for all x.
So there is a derivative and it must be constant. Since f(0) = 2f(0) the function must be ax for some a.
HOW do you assert "so there is a derivative"? There is a derivative if and only if [itex]\lim_{h\to 0} f(h)/h[/itex] exists but how do you show that?
i wasn't asserting anything - just suggesting a line of argument.
I guess at some point you have to use continuity. look at expressions like f(h/n)/h/n = f(h)/h so the ratio is constant for rationals. Then use continuity.
The thing is that without continuity the assertion is false.
Wasn't sure about the Newton quotient though...
Suggested for: Examples which satistfy f(x+y)=f(x) + f(y)
- Last Post
- Last Post
- Last Post
- Last Post
- Last Post
- Last Post
- Last Post
- Last Post
- Last Post
- Last Post
Source: https://www.physicsforums.com/threads/examples-which-satistfy-f-x-y-f-x-f-y.486409/
0 Response to "Prove That if F is Continuous on r and F X Y F X F Y for Then F X Ax for Some"
Post a Comment