Proving f is Continuous Everywhere: Spivak Calc Problem

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In summary, the definition of "continuous everywhere" for a function is that for every point a in the domain of f, the limit of f(x) as x approaches a is equal to f(a). To prove a function is continuous everywhere, one must show that the limit of f(x) as x approaches a is equal to f(a) for every point a in the domain of f, using the epsilon-delta definition of a limit. Proving a function is continuous everywhere is important because it ensures the function's well-defined behavior and allows for the use of powerful tools for analysis. An example of a function that is not continuous everywhere is the piecewise function f(x) = {1, if x is rational; 0, if
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cesc
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Homework Statement



f is a function that satisfies
f(x+y)=f(x)+f(y) and f is continuous at 0.

prove f is continuous everywhere


Homework Equations





The Attempt at a Solution


its easy to see that f(0)=0

My hunch is that the only soln f= cx, and f=0;
but otherwise can't make much headway
 
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Write down the definition of continuous at x=0 using f(0)=0. Substitute x-a for x. Can you change it into the definition of f being continuous at x=a?
 

Related to Proving f is Continuous Everywhere: Spivak Calc Problem

1. What is the definition of "continuous everywhere" for a function?

The definition of "continuous everywhere" for a function f is that for every point a in the domain of f, the limit of f(x) as x approaches a is equal to f(a). In other words, the function has no abrupt changes or jumps in its graph.

2. How do you prove that a function f is continuous everywhere?

To prove that a function f is continuous everywhere, you need to show that the limit of f(x) as x approaches a is equal to f(a) for every point a in the domain of f. This can be done using the epsilon-delta definition of a limit, which states that for any given epsilon, there exists a delta such that if the distance between x and a is less than delta, then the distance between f(x) and f(a) is less than epsilon.

3. What is the importance of proving a function is continuous everywhere?

Proving that a function is continuous everywhere is important because it ensures that the function is well-defined and behaves in a predictable manner. It also allows us to use powerful tools such as the Intermediate Value Theorem and the Extreme Value Theorem to analyze the behavior of the function.

4. Can you give an example of a function that is not continuous everywhere?

Yes, a classic example of a function that is not continuous everywhere is the piecewise function f(x) = {1, if x is rational; 0, if x is irrational}. This function is not continuous at any point, as there is a jump in its graph at every point in its domain.

5. Are there any shortcuts or tricks to proving a function is continuous everywhere?

No, there are no shortcuts or tricks to proving a function is continuous everywhere. It requires a rigorous application of the epsilon-delta definition of a limit and careful analysis of the function's behavior at every point in its domain. However, with practice, the process becomes more intuitive and easier to understand.

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