# Real analysis

Not what you're looking for?

Let f be a function defined on all of R that satisfies the additive condition f(x+y)=f(x)+f(y) for all x,y belong to R

a- Show that f(0)=0 and that f(-x)=-f(x) for all x belong to R.

b- Show that if f is continuous at x=0 then f is continuous at every point in R

c- Let k=f(1) show that f f(n)=kn for all n belong to N and then prove that f(z)=kz for all z belong to Z. now prove that f(r)=kr for any rational number r.

##### Purchase this Solution

##### Solution Summary

The solution is a proof regarding continuous functions.

##### Solution Preview

Proof:

a. f(0+x)=f(0)+f(x), then we have f(x)=f(0)+f(x). So f(0)=0

f(0)=f(-x+x)=f(-x)+f(x)=0, thus f(-x)=-f(x)

b. Suppose f is continous at 0, then for any e>0, there exists some d>0, such that when |x|<d, ...

##### Purchase this Solution

##### Free BrainMass Quizzes

##### Probability Quiz

Some questions on probability

##### Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

##### Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

##### Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

##### Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.