Register to reply

How to solve this parameter proof

by transgalactic
Tags: parameter, proof, solve
Share this thread:
transgalactic
#1
Mar13-09, 12:45 PM
P: 1,397
f(x) is a continues function on (-infinity,+infinity) for which
f(x+y)=f(x)+f(y)

prove that there is parameter a for which f(x)=ax for every real x

i was given a hint to solve it for x in Q

there is no much thing i can do here for which i can use theorems

the only thing i am given that its continues

lim f(x)=f(x)

??
Phys.Org News Partner Science news on Phys.org
Physical constant is constant even in strong gravitational fields
Montreal VR headset team turns to crowdfunding for Totem
Researchers study vital 'on/off switches' that control when bacteria turn deadly
HallsofIvy
#2
Mar13-09, 01:35 PM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,683
1. Prove by induction that f(nx)= nf(x) for any positive integer n and any real number x.

2. From that, taking x= 0, show that f(0)= 0.
From here on, n will represent any integer and x any real number.

3. Prove, by looking at f((n+(-n))x), that f(-nx)= -f(nx).

3. Prove, by looking at f(n(x/n)), that f(x/n)= f(x)/n.

4. Prove that, for any rational number, r, f(r)= rf(1).

5. Use the continuity of f to show that f(x)= xf(1) for any real number, x.
transgalactic
#3
Mar13-09, 02:54 PM
P: 1,397
regarding 1:
f(kx)=kf(x) given
prove f(kx + x)=(k+1)f(x)

i dont know how to use the given
??

Mark44
#4
Mar13-09, 02:59 PM
Mentor
P: 21,409
How to solve this parameter proof

f(kx + x) = f(kx) + f(x) = kf(x) + f(x) = (k + 1)f(x)
The first step of the chain of equality above comes from the assumption in the original problem.


Register to reply

Related Discussions
How to use mean theorem to solve this proof Calculus & Beyond Homework 4
Solve this Proof Calculus & Beyond Homework 2
One- parameter groups Differential Geometry 3
Limit with parameter Calculus 3
Which parameter. General Math 0