# Proof a function is continuous

• Dr. Science
In summary, the statement presents a function f with the property that the absolute value of the difference between f(x) and f(t) is less than or equal to the absolute value of the difference between x and t for any pair of points in the interval (a,b). The task is to prove that f is continuous on the same interval and the attempt at a solution suggests that f(x) and f(t) must be defined everywhere on (a,b). The conversation continues with a discussion of the definition of continuity at a point and the need for an epsilon-delta proof without a specific Homework Equation provided.
Dr. Science

## Homework Statement

Suppose that a function f has the property that

|f(x) - f(t)| < or = |x-t| for each pair of points in the interval (a,b). Prove that f is continuous
on (a,b)

?

## The Attempt at a Solution

f(x) and f(t) must be defined everywhere on (a,b)

Dr. Science said:

## Homework Statement

Suppose that a function f has the property that

|f(x) - f(t)| < or = |x-t| for each pair of points in the interval (a,b). Prove that f is continuous
on (a,b)

?

## The Attempt at a Solution

f(x) and f(t) must be defined everywhere on (a,b)

By "prove" do you mean an epsilon-delta proof?

i have no idea, that's all the question says. it just says to prove f is continuous

What's the definition of continuity at a point?

as x approaches c f(x) = f(c)

Write down a more precise definition and then pick epsilon appropriately; make it slightly greater than delta. :-)

## 1. What does it mean for a function to be continuous?

A function is continuous if it has no breaks or gaps in its graph. This means that the values of the function change gradually and smoothly as the input values change.

## 2. How can you prove that a function is continuous?

To prove that a function is continuous, you can use the epsilon-delta definition of continuity. This involves showing that for any given epsilon (a small positive number), there exists a delta (a small positive number) such that for any x within delta units of a given value, the difference between the function values at x and the given value is less than epsilon. If this condition is satisfied, the function is considered continuous at that point.

## 3. What is the importance of continuity in mathematics and science?

Continuity is important because it allows us to make predictions and draw conclusions about a function even when we only have information about its behavior at a limited number of points. It also allows us to apply calculus to the function and use techniques such as differentiation and integration.

## 4. Can a function be continuous at some points but not others?

Yes, a function can be continuous at some points but not others. This is known as a point discontinuity and occurs when the function has a break or gap at a specific point. For example, a function may be continuous everywhere except at x = 0, where it has a vertical asymptote.

## 5. How does the continuity of a function relate to its differentiability?

If a function is continuous at a point, it does not necessarily mean that it is differentiable at that point. However, if a function is differentiable at a point, it must also be continuous at that point. This means that differentiability is a stronger condition than continuity.

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