# Proving Property of a Continuous Function

• xx0x0xx
In summary, the conversation discusses the use of the epsilon delta definition of continuity in solving a problem regarding intervals and determining the value of f(v). The conversation includes clarifications on the reasoning and steps involved in the solution process.
xx0x0xx

Continuity @ v0

## The Attempt at a Solution

Using the epsilon delta definition of continuity:

If we choose epsilon such that epsilon < a, then |f(v) - f(v0)| < a.
So f(v) is in the interval (f(v0) - a, f(v0) + a).
Only half of this interval is what I want though.
I may be doing the whole thing wrong… I would really appreciate any help

edit: Actually, I'm pretty sure my reasoning is completely wrong, so I'll think about it some more...

Last edited:
i think you're on the right track, so just clarifying what you've got:

you know f(v0)>a, so define M by f(v0)-a=M>0

now use the continuity of f and choose e>0, such that

|f(v0)-f(v)|<e<M

then there exist d>0 such that for |v0-v|<d, then |f(v0)-f(v)|<M/2

if f(v) > f(v0) we're fine, if f(v)< f(v0) then

|f(v0)-f(v)| = f(v0)-f(v) < M = f(v0) -a

then
f(v) > a

(note drawing a picture really helps for this one)

## What is a continuous function?

A continuous function is a mathematical function where the output changes smoothly and continuously as the input changes. This means that there are no sudden jumps or breaks in the graph of the function.

## How do you prove that a function is continuous?

To prove that a function is continuous, you must use the epsilon-delta definition of continuity. This involves showing that for any arbitrarily small number (epsilon), there exists a corresponding small interval (delta) around the input value where the output values of the function remain within a certain range. If this condition is met, the function is considered continuous.

## What are some common properties of continuous functions?

Some common properties of continuous functions include: the graph of the function is a single, unbroken curve, the function has a limit at every point, the function can be evaluated at any point within its domain, and the function can be integrated and differentiated at any point within its domain.

## Can a function be continuous at one point but not at another?

Yes, it is possible for a function to be continuous at one point but not at another. This occurs when the function has a jump or discontinuity at a specific point. However, the function can still be considered continuous if the jump is small enough and can be "smoothed out" by connecting the points on either side of the jump.

## What is the importance of proving the property of continuity in a function?

Proving the property of continuity in a function is important because it ensures that the function is well-defined and behaves predictably. It also allows us to use important mathematical tools such as the Intermediate Value Theorem and the Mean Value Theorem, which rely on the concept of continuity.

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