What does a function takes on mean?

In summary, "a function takes on" means that the function's range or image includes all possible values, except for possibly one "exceptional value". This exceptional value would make the statement false, indicating that the function does not take on that particular value.
  • #1
Nikitin
735
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What does "a function takes on" mean?

Hey! I'm currently learning Picard's theorem, but I don't quite understand it because of difficult words..

In the text, it uses the term "takes on", ie

"If f(z) is analytic and has an isolated essential singularity at a point z_0, it takes on every value, with at most one exceptional value in an arbitrarily small ε-neighbourhood of z_0".

What does "take on" mean? Are they talking about the function's range? And what do they mean with "exceptional value"?
 
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  • #2
Nikitin said:
What does "take on" mean? Are they talking about the function's range?
Yes - or technically the "image" of the function. http://en.wikipedia.org/wiki/Codomain

And what do they mean with "exceptional value"?

It means a complex number c such that y=c is an exception to the statement "the function's image contains the value y". That is, y = c makes the statement false.
 

FAQ: What does a function takes on mean?

1. What is the definition of "takes on" in relation to a function?

The phrase "takes on" is often used to describe the output or value of a function. It refers to the process of inputting a specific value into a function and obtaining a corresponding output.

2. How is "takes on" different from "gives" in terms of a function?

While "takes on" describes the output of a function, "gives" is used to describe the relationship between the input and output. A function takes on a value, while it gives a relationship between two values.

3. Can a function take on multiple values?

Yes, a function can take on multiple values. This is known as a many-to-one function, where multiple inputs can produce the same output. For example, the square root function can take on both positive and negative values for a given input.

4. Is "takes on" the same as "returns" in relation to a function?

Yes, "takes on" and "returns" can be used interchangeably when discussing the output of a function. Both phrases refer to the value that a function produces when given a specific input.

5. How does "takes on" relate to the range of a function?

The range of a function describes all the possible output values that a function can take on. Therefore, "takes on" is often used to discuss the range of a function, as it describes the specific values that a function can produce.

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