Prove Continuity From Precise Definition of Limit

[toslowtogofast2a]

Homework Statement

The problem tells us f is continuous at 0 and that if f(a+b) = f(a)+f(b) then prove f is continuous at every number.

Relevant Equations

The solution in the book used a different approach but I am trying to start with the precise definition of continuity and prove from there.

For all epsilon >0 there exists delta >0 ST
|f(x)-f(c)|<epsilon. When. 0<|x-c|<delta

I attached my attemp at the solution. I am trying to start with continuity at 0 and end up with limit of f(x) equals f(c) as x goes to c.

Could someone take a look at the attached image and let me know if I am on the right track or where I went astray

Sorry picture is rotated I tried but can’t get it to come in right.

 

[FactChecker]

Although your description of your proof is straightforward, the organization of your proof is confusing. It's hard to see if it is correct.
It's not clear that it is following your description: "I am trying to start with continuity at 0 and end up with limit of f(x) equals f(c) as x goes to c."
1) You should start in a way that briefly makes it clear which point continuity is being established (i.e. define $c\in\mathbb{R}$). Also define $\epsilon\gt 0$.
2) Use the continuity of $f()$ at 0 to get $\delta$:
3) Use that $\delta$ to prove continuity at $c$:

 

[PeroK]

Note that there is an alternative, equivalent formulation of continuity using sequences, which is often useful for proofs. It's fairly to simple to show that $f$ is continuous at $x$ iff for every sequence $x_n$ that converges to $x$, the sequence $f(x_n)$ converges to $f(x)$. I'm not sure why this is not taught more widely.

Anyway, in this case, you could use that alternative definition for a nice, simple proof.

 

[toslowtogofast2a]

Although your description of your proof is straightforward, the organization of your proof is confusing. It's hard to see if it is correct.
It's not clear that it is following your description: "I am trying to start with continuity at 0 and end up with limit of f(x) equals f(c) as x goes to c."
1) You should start in a way that briefly makes it clear which point continuity is being established (i.e. define $c\in\mathbb{R}$). Also define $\epsilon\gt 0$.
2) Use the continuity of $f()$ at 0 to get $\delta$:
3) Use that $\delta$ to prove continuity at $c$:

@FactChecker Thanks for the reply. I agree I let out details and was unclear. I tried to be more detailed and clear about my structure in image below. Is this better.

 

[FactChecker]

That looks good to me. I might quibble about some "wordsmithing" things, but the logic and flow seems fine.

 

[toslowtogofast2a]

That looks good to me. I might quibble about some "wordsmithing" things, but the logic and flow seems fine.

Thanks for taking the time to help me with this

 

[PeroK]

Just to show the alternative proof. Let $x \in \mathbb R$.

Let $x_n$ be a sequence that converges to $x$. Then, $x_n - x$ is a sequence that converges to $0$. As $f$ is continuous at $0$, $f(x_n - x)$ converges to $f(0)$. Using the linear property of $f$, we have $f(x_n - x) = f(x_n) - f(x)$. Hence $f(x_n)$ converges to $f(0) + f(x) = f(0 + x) = f(x)$.

This shows that $f(x_n)$ converges to $f(x)$, hence $f$ is continuous at $x$.

 

[toslowtogofast2a]

Just to show the alternative proof. Let $x \in \mathbb R$.

Let $x_n$ be a sequence that converges to $x$. Then, $x_n - x$ is a sequence that converges to $0$. As $f$ is continuous at $0$, $f(x_n - x)$ converges to $f(0)$. Using the linear property of $f$, we have $f(x_n - x) = f(x_n) - f(x)$. Hence $f(x_n)$ converges to $f(0) + f(x) = f(0 + x) = f(x)$.

This shows that $f(x_n)$ converges to $f(x)$, hence $f$ is continuous at $x$.

Thanks for posting that. I have not learned that way yet but it is a good tool to have in my bag. It is a nice solution to the problem and pretty concise