Is the Function Defined by the Infimum of Distances Continuous?

In summary, the conversation discusses the continuity of a function in a metric space, defined by taking the infimum of the distance between a point in the space and a nonempty subset. The conversation also suggests two approaches for proving the continuity of this function, one using the abstract definition and the other using the epsilon-delta definition. The conversation ends with a proposed inequality to prove the continuity of the function.
  • #1
tylerc1991
166
0

Homework Statement



Let (X,d) be a metric space and let A be a nonempty subset of X. Define a function f:X -> R^1 by f(x) = inf{d(x,a) : a is an element of A}. Prove that f is continuous.

Homework Equations





The Attempt at a Solution



Intuitively I can see that the function is continuous because it seems like for an arbitrary open interval in R^1 there is some pre-image of the function that is an open subset of this open interval, I just don't exactly know where to begin writing this. Can someone help me with the intuition behind this problem and let me know if I am on the right track? Thank you very much!
 
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  • #2
If you don't know how to attack a continuity problem using the abstract (preimage of an open set is open) definition, try instead to prove that the function is continuous at any arbitrarily chosen point using the good old epsilon-delta definition.
 
  • #3
In both cases, you may want to prove that
[tex]|f(x)-f(y)|\leq d(x, y)[/tex]
 

FAQ: Is the Function Defined by the Infimum of Distances Continuous?

What is topology continuity?

Topology continuity is a fundamental concept in mathematics that deals with the connectedness and smoothness of a mathematical space. It is used to determine if a function or transformation preserves the structure or properties of a given space.

What is a topology continuity proof?

A topology continuity proof is a mathematical proof that shows a function or transformation is continuous in a given topological space. It involves using the definition of continuity and the properties of the space to demonstrate that the function preserves the structure of the space.

What are the steps involved in a topology continuity proof?

The steps involved in a topology continuity proof are as follows:
1. Write out the definition of continuity for the given space.
2. Show that the function satisfies the definition of continuity.
3. Use the properties of the space to demonstrate that the function preserves the structure of the space.
4. Provide a logical argument to show that the function is continuous in the given space.
5. Conclude the proof by summarizing the steps and stating that the function is continuous in the given space.

What are some common mistakes to avoid in a topology continuity proof?

Some common mistakes to avoid in a topology continuity proof include:
- Not understanding the definition of continuity and using it incorrectly.
- Not using the properties of the space to demonstrate the preservation of structure.
- Skipping steps or not providing a logical argument for the proof.
- Confusing topology continuity with other types of continuity.
- Not being familiar with the properties of the given space.

Why is topology continuity important?

Topology continuity is important because it allows us to study and understand the behavior of functions and transformations in a given space. It also helps us to determine if a function is continuous and if it preserves the structure of the space. This is crucial in many areas of mathematics and science, such as in analysis, differential equations, and physics.

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