Prove Continuity at a: f(x+y)=f(x)+f(y) & f(0) Continuous

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In summary, by using the given information that f(x+y)=f(x)+f(y) and that f is continuous at 0, we can prove that f is also continuous at any arbitrary point a. This is done by reducing the problem around a to a problem around 0, using the properties of continuity and the given equation. This proves that f is continuous for all values of a.
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Homework Statement

Suppose that "f" satisfies "f(x+y)=f(x)+f(y)", and that "f" is continuous at 0. Prove that "f" is continuous at a for all a.

Homework Equations

In class we were given 3 main ways to solve continuity proofs.

A function "f" is continuous at x=a if:

a.)
Limit of f(x) as x->a = f(a), (a can be all real numbers)

b.)
Limit of f(a+h) as h->0 = f(a), (Let x=a+h)

c.)
(delta-epsilon proof)
For all epsilon greater than 0, there exists some delta greater than 0, such that for all x, if |x-a|< delta then |f(x)-f(a)|< epsilon.

The Attempt at a Solution

So far, by working with my teacher I was able to get this much as being correct:

Theres not enough information for a delta-epsilon proof so

[
Were told that f(x+y)=f(x)+f(y), for all x,y belonging to real numbers then
In partiality:
f(0+0)=f(0)+f(0)
f(0)=2f(0)
Only way this can be possible is if f(0)=0
hence f(0)=0

Now consider:

Limit of f(a+h)-f(a) as h->0 is equal to 0.
]

Im not as to what i should do now and what the second part that my teacher wants me to consider will prove.

Any ideas would be greatly appreciated!

Use the information given. You can rewrite f(a + h) - f(a) to reduce the problem around a to a problem around zero.

How can I reduce the problem to be around zero?
Also what does this prove?

OK, firstly: do you see what $$\lim_{h \to 0} f(a + h) - f(a) = 0$$ is good for?

Then, note that you have been given that f is continuous at zero and you want to prove continuity at some arbitrary a, about which you don't know anything. So you can try to use the properties you have to make the problem of continuity at a into a problem of continuity at 0, because you have more information of f around 0 than around a.

You have to use the fact that f(x+ y)= f(x)+ f(y). So f(a+ h)= ?

1. What does it mean for a function to be continuous at a point?

Continuity at a point means that the function is defined and has a value at that point, and as the input approaches that point, the output of the function also approaches the same value.

2. How is continuity at a point determined mathematically?

A function is continuous at a point if the limit of the function as x approaches that point is equal to the value of the function at that point.

3. How does the given equation f(x+y)=f(x)+f(y) relate to continuity at a point?

The given equation is known as the "additivity property" and it is a necessary condition for a function to be continuous at a point. It means that the value of the function at a point is equal to the sum of the values of the function at two separate points.

4. How is continuity at a point different from continuity on an interval?

Continuity at a point only considers the behavior of the function at a single point, while continuity on an interval considers the behavior of the function on a range of values. A function can be continuous at a point but not on an interval, and vice versa.

5. What is the significance of f(0) being continuous in the given equation?

Having f(0) be continuous means that the function has a defined value at the point 0, and the function maintains this property even as the input approaches 0. This is an important condition for the function to be continuous at 0.

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