Proof a function is continuous

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.
  • #1
Dr. Science
31
0

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)

Homework Equations



?

The Attempt at a Solution



f(x) and f(t) must be defined everywhere on (a,b)
 
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  • #2
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)

Homework Equations



?

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?
 
  • #3
i have no idea, that's all the question says. it just says to prove f is continuous
 
  • #4
What's the definition of continuity at a point?
 
  • #5
as x approaches c f(x) = f(c)
 
  • #6
Write down a more precise definition and then pick epsilon appropriately; make it slightly greater than delta. :-)
 

Related to Proof a function is continuous

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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