1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
larisa [96]
3 years ago
10

Suppose that f: R --> R is a continuous function such that f(x +y) = f(x)+ f(y) for all x, yER Prove that there exists KeR su

ch that f(x) = kx, for every XER
Mathematics
1 answer:
Pachacha [2.7K]3 years ago
4 0
<h2>Answer with explanation:</h2>

It is given that:

f: R → R is a continuous function such that:

f(x+y)=f(x)+f(y)------(1)  ∀  x,y ∈ R

Now, let us assume f(1)=k

Also,

  • f(0)=0

(  Since,

f(0)=f(0+0)

i.e.

f(0)=f(0)+f(0)

By using property (1)

Also,

f(0)=2f(0)

i.e.

2f(0)-f(0)=0

i.e.

f(0)=0  )

Also,

  • f(2)=f(1+1)

i.e.

f(2)=f(1)+f(1)         ( By using property (1) )

i.e.

f(2)=2f(1)

i.e.

f(2)=2k

  • Similarly for any m ∈ N

f(m)=f(1+1+1+...+1)

i.e.

f(m)=f(1)+f(1)+f(1)+.......+f(1) (m times)

i.e.

f(m)=mf(1)

i.e.

f(m)=mk

Now,

f(1)=f(\dfrac{1}{n}+\dfrac{1}{n}+.......+\dfrac{1}{n})=f(\dfrac{1}{n})+f(\dfrac{1}{n})+....+f(\dfrac{1}{n})\\\\\\i.e.\\\\\\f(\dfrac{1}{n}+\dfrac{1}{n}+.......+\dfrac{1}{n})=nf(\dfrac{1}{n})=f(1)=k\\\\\\i.e.\\\\\\f(\dfrac{1}{n})=k\cdot \dfrac{1}{n}

Also,

  • when x∈ Q

i.e.  x=\dfrac{p}{q}

Then,

f(\dfrac{p}{q})=f(\dfrac{1}{q})+f(\dfrac{1}{q})+.....+f(\dfrac{1}{q})=pf(\dfrac{1}{q})\\\\i.e.\\\\f(\dfrac{p}{q})=p\dfrac{k}{q}\\\\i.e.\\\\f(\dfrac{p}{q})=k\dfrac{p}{q}\\\\i.e.\\\\f(x)=kx\ for\ all\ x\ belongs\ to\ Q

(

Now, as we know that:

Q is dense in R.

so Э x∈ Q' such that Э a seq belonging to Q such that:

\to x )

Now, we know that: Q'=R

This means that:

Э α ∈ R

such that Э sequence a_n such that:

a_n\ belongs\ to\ Q

and

a_n\to \alpha

f(a_n)=ka_n

( since a_n belongs to Q )

Let f is continuous at x=α

This means that:

f(a_n)\to f(\alpha)\\\\i.e.\\\\k\cdot a_n\to f(\alpha)\\\\Also\\\\k\cdot a_n\to k\alpha

This means that:

f(\alpha)=k\alpha

                       This means that:

                    f(x)=kx for every x∈ R

You might be interested in
Jasmine took a cab home from her office.the cab charged a flat fee of $4 plus $3 per mile jasmine $32 for the tip. how far does
Annette [7]
32 = 4+3x
X= 9.3 miles away from the office
7 0
3 years ago
Brian invests £600 into his bank account.
Savatey [412]

Answer:

It will be £2520

Step-by-step explanation:

  1. 3.2=
  2. \frac{16}{5}
  3. \frac{16}{5}  \times 100 = 320
  4. 320
  5. 320 \times 6 = 1920
  6. 1920 + 600 = 2520
7 0
3 years ago
Which ordered pair is a solution of the equation 4x - 2y = 20? *
xxTIMURxx [149]

Answer:

(6,2)

Step-by-step explanation:

Start by plugging in 6 for x and 2 for y

4(6)-2(2) = ?

24-4 = 20

4 0
3 years ago
How would you graph the equation above y=3x+3 on a coordinate plane?
Trava [24]

Answer:

You would start at 3 on the y-axis and go up 1 and right 3

Step-by-step explanation:

6 0
3 years ago
What is the approximate difference in the growth rate of the two populations? The approximate difference in the growth rate is 1
motikmotik
B IS THE ANSWER HOPE THIS HELPS
5 0
3 years ago
Read 2 more answers
Other questions:
  • What is the value of x?<br><br> A. 90<br> B. 95<br> C. 100<br> D. 115
    13·1 answer
  • 9. Find the area of each figure to the nearest tenth.
    10·2 answers
  • Please help as soon as possible
    5·1 answer
  • Describe possible consequences of each of the following.
    8·1 answer
  • Bianca earned $399.60 after working for 18.5 hours. How much did she earn for one hour?​
    7·2 answers
  • Adric has a rope 9 yards long. He cuts off 7 pieces of rope,
    14·1 answer
  • Plz help me I need to pass this test
    15·1 answer
  • Find the value of x.
    10·1 answer
  • The Meyer family spent $65 on dinner and left a 20% tip. What was the total cost of their meal?
    5·1 answer
  • Joe has a 300 foot fence around his rectangular yard. The length is 10 feet more than the width. Which equation can you use to d
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!