Continuity of a Function with Inverse Preimage Condition

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In summary, the conversation discusses the implications of a function f being continuous if for each open set U in Y containing some element of f(X), f^-1(U) is open in X. It is concluded that this is true because if U does not contain any element of f(X), f^-1(U) would be empty, which is still considered an open set. The conversation also mentions that this version of continuity is equivalent to other definitions of continuity, such as the epsilon-delta definition.
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fraggle
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Homework Statement



Suppose f:X-->Y
suppose for each open set U in Y s.t U contains some element f(x), we have f^(-1)(U) is open in X.
Does this imply f is continuous

Homework Equations


U is not quite an arbitrary open set of Y since there could be an open set of Y that does not interesct the image f(X).
Suppose V is such a set.
Then can we say f^(-1)(V)= the empty set? Or is it just undefined?

This isn't a homework question, just curious.



The Attempt at a Solution


I believe it is continuous since, otherwise the proof I'm reading in a textbook wouldn't work (they just ommited this part).

Could anyone explain the reasoning for this?
thank you
 
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  • #2
You're right since if the open set U didn't contain the image of any point in X, then f-1(U) would be empty (since nothing in X was mapped to U), and the empty set is an open set. So they are just excluding that case since it doesn't really carry any information with it.

Also, this version of continuity is equivalent to the epsilon-delta definition of continuity, as well as a few other versions. For example, if for every closed set in Y, f inverse of that closed set is closed in X, then f is continuous. You can prove it from the above statement by taking complements.
 

FAQ: Continuity of a Function with Inverse Preimage Condition

What is simple continuity?

Simple continuity is a concept in mathematics and physics that refers to the unbroken or uninterrupted nature of a function or physical system. It means that the values of the function or properties of the system do not suddenly change or jump to a different value without any logical explanation.

What is the difference between simple continuity and uniform continuity?

Simple continuity refers to the overall smoothness and unbroken nature of a function or system, while uniform continuity specifically refers to the behavior of a function or system over a specific interval. Uniform continuity requires that the function or system maintains a consistent rate of change over that interval.

How is simple continuity different from differentiability?

Simple continuity is a necessary condition for differentiability, but it is not sufficient. A function or system can be continuous but not differentiable if there are sharp corners or discontinuities in the function. Differentiability requires that the function or system is continuous and has a defined slope at every point.

What are some real-life examples of simple continuity?

Simple continuity can be observed in many natural and man-made systems. Examples include the smooth flow of water in a river, the gradual change in temperature throughout the day, and the consistent acceleration of a car on a straight road. It can also be seen in mathematical functions, such as a line or a parabola.

How is simple continuity used in science and engineering?

Simple continuity is a fundamental concept in many branches of science and engineering. It is used to describe and analyze the behavior of physical systems, from simple motion to complex systems like fluid dynamics. In mathematics, it is used to prove the existence of solutions to equations and to establish the continuity of functions.

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