Prove Continuity of f at a w/ f(x+y)=f(x)+f(y)

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In summary, the book solution provides a proof that f is continuous at a for all a if you redefine the variables and show that lim_{x-->a}[f(x)]=f(a).
  • #1
freshlikeuhh
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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



f(x+y) = f(x) + f(y)
Limit Definition
Continuity: f is continuous at a if the limit as x approaches a is the value of the function at a.

The Attempt at a Solution



I am not even sure how to approach this question. I have already seen a solution to this question but I do not understand it. That solution is as follows:

Note that f(x+0) = f(x) + f(0), so f(0)=0. Now:

(h->0)lim f(a+h) - f(a) = (h->0)lim f(a) + f(h) - f(a) = (h->0)lim f(h) = (h->0) f(h) - f(0) = 0, since f is continuous at 0.

This is in the back of my textbook and I don't understand how that concludes anything.
 
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  • #2
It may help to state what you wish to show explicitly. Based on the definition of continuity, you want to show that lim_{x-->a}[f(x)]=f(a) (at the same time you'll verify that the limit exists). Can you see how to redefine variables and rewrite the expression above so that this expression is what the book solution is verifying (though the last step before finding "0" seems superfluous)?
 
  • #3
javierR said:
It may help to state what you wish to show explicitly. Based on the definition of continuity, you want to show that lim_{x-->a}[f(x)]=f(a) (at the same time you'll verify that the limit exists). Can you see how to redefine variables and rewrite the expression above so that this expression is what the book solution is verifying (though the last step before finding "0" seems superfluous)?

Well, I understand how the property f(x+y)=f(x)+f(y) is used to demonstrate f(0)=0 and obviously it is employed to expand f(a+h)=f(a)+f(h). I understand that this step is supposed to demonstrate the continuity of all a, but then why is h->0, and how exactly is continuity of a demonstrated if a is canceled out? Is this just using that fact that f(a - a) = f(0), which is 0?

What would an alternate solution look like?
 

Related to Prove Continuity of f at a w/ f(x+y)=f(x)+f(y)

1. What is continuity?

Continuity refers to the property of a mathematical function where small changes in the input result in small changes in the output. In other words, a function is continuous if it has no sudden jumps or breaks in its graph.

2. What does it mean to prove continuity?

Proving continuity involves showing that a function satisfies the definition of continuity at a specific point or over a specific interval. This typically involves using mathematical techniques such as the epsilon-delta method or the sequential criterion for continuity.

3. How do you prove continuity of a function at a specific point?

In order to prove continuity at a specific point, you must show that the limit of the function at that point exists and is equal to the value of the function at that point. This can be done using the epsilon-delta method, where you choose a small value for epsilon and then find a corresponding delta that satisfies the definition of continuity.

4. What is the sequential criterion for continuity?

The sequential criterion for continuity states that a function is continuous at a point if and only if the limit of the function at that point is equal to the value of the function at that point for every sequence of values that converges to that point. In other words, if a function passes the sequential test, it is continuous at that point.

5. How do you prove continuity of a function over an interval?

In order to prove continuity over an interval, you must show that the function is continuous at every point within that interval. This can be done by using the sequential criterion for continuity or by breaking the interval into smaller sub-intervals and proving continuity at each one using the epsilon-delta method.

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