# Proving f is Continuous Everywhere: Spivak Calc Problem

• cesc
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
cesc

## 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

## 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

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?

## 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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