limit and derivative

Question

if f(x+y) = f(x) + f(y), ∀ x,y ∈ R and f is continuous at one point of R , then f is/are

 

a) not continous ∀ R

b) uniformly continous ∀ R

c) differentiable ∀ R

d) none of these

 

please explain it, please help me sir plz

Answer 1

First of all, plugging in x=0 gives you that f(y)=f(0)+f(y) or f(0)=0.

 

Secondly, say that f(1) = s. Using induction it follows that f(n)=ns.

 

Say also that f(a) = t for some rational a ∉ {-1,0,1}. Then f(na) = nt by induction.

 

It also follows that s=t since some integer multiple of a gives an integer.

 

This shows that f(x) = sx for all rational x.

 

Now let y be an irrational numberand q be rational. Then f(q+y)=f(q)+f(y) = f(y)+sq

 

If f(y) ≠ ys then because the set Q of rational numbers is everywhere dense in R, f(x) would be nowhere continuous.

 

However, we are given it is somewhere continuous. So we are forced to conclude that the family of functions satisfying f(x+y) = f(x)+f(y) for all x,y ∈ R are exactly those where there is a constant s ∈ R such that f(x) = sx for all x ∈ R.

 

This conclution makes it continuous in all R, so you can eliminate (a).

It is uniformally continuous, just take δ such that |s|δ = ε for all ε>0 (s≠0) or any δ (s=0) as |f(y)-f(x)| = s|y-x|.

It is differentiable so you get that the answers are (b) and (c).