Function that equals 1 at x=0, but 0 everywhere else?

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In summary, the conversation discusses various functions that equal 1 at x=0 and 0 everywhere else without using a piecewise function. The suggestion of taking the limit of a sequence of functions is explored, with the final suggestion being f(x) = 0^(x^2), which is modified to f(x) = 0^(x^2) to avoid issues with negative real numbers. The topic of whether 0^0 equals 1 or is an indeterminate form is briefly mentioned but not further discussed.
  • #1
Kepler_
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Is there a function that equals 1 at x=0 and equals 0 when x isn't 0 without using a piecewise function? I've been experimenting with limits and derivatives but haven't made much progress.

The closest thing I was able to think of is 1-x/x, which is indeterminate at x=0, and 0 everywhere else.

Thanks!
 
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  • #2
How about the limit as [itex]a[/itex] approaches [itex]\infty[/itex] of [itex]f(x) = e^{-a x^{2}}[/itex]?
 
  • #3
That'll do it :P
It seems like any real constant greater than 1 works in place of e. Is there a reason why you chose e?
 
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  • #4
Kepler_ said:
That'll do it :P
It seems like any non-zero real constant works in place of e. Is there a reason why you chose e?

Out of infinitely many possible bases to choose from, ##e## is the only one that makes sense most of the time. In this case it doesn't matter, but why would you choose something else? If you want, you could rewrite this as
##
\lim_{a\rightarrow\infty}\left(\frac{1}{e^a}\right)^{x^2}
##in which case it is the same as
##
\lim_{b\rightarrow0}\left( b\right)^{x^2}
##
What matters is the ##x^2## exponent and the continuous, decreasing function as a base. Most mathematicians are used to ##e## being part of that base.

Also, this is sometimes thought of as a limit of normal distributions. These are defined using ##e##.
 
  • #5
without using a piecewise function?

Is there a reason for this requirement? "Piecewise" is just a notation for describing a function; it isn't actually a "kind" of function. Even your description of your function is piecewise: 1 at zero, and 0 everywhere else.
 
  • #6
Two functions are equal if they take the same value at every point. So, regardless of whether you describe a function piecewise or as the limit of a sequence of functions, it's the same function in the end.
 
  • #7
I think this demonstrates a fundamental misunderstanding of functions. Functions are a rule associating two sets. That's it. Really, "piecewise function" doesn't really make sense. It's just a way of writing things.
 
  • #8
johnqwertyful said:
I think this demonstrates a fundamental misunderstanding of functions. Functions are a rule associating two sets. That's it. Really, "piecewise function" doesn't really make sense. It's just a way of writing things.

But sometimes piecewise makes calculus hard. I think this is a good point to make though.
 
  • #9
DrewD said:
But sometimes piecewise makes calculus hard. I think this is a good point to make though.

But "Piecewise" is only a way of describing a function. There is no such thing as a "piecewise function". f(x)=x if x>0, -x if x≤0. Or f(x)=|x|. Both are the same function. One is written "piecewise" the other isn't.
 
  • #10
johnqwertyful said:
But "Piecewise" is only a way of describing a function. There is no such thing as a "piecewise function". f(x)=x if x>0, -x if x≤0. Or f(x)=|x|. Both are the same function. One is written "piecewise" the other isn't.

Agreed.
 
  • #11
Here's another function that would satisfy the requirement but no limits involved:

f(x) = 0^x
 
  • #12
paisiello2 said:
Here's another function that would satisfy the requirement but no limits involved:

f(x) = 0^x

How would that be defined for negative real numbers?
 
  • #13
Ooops, you're right, I forgot to modify it:

f(x) = 0^(x^2)
 
  • #14
Everytime I have encountered it, ##0^0## has been considered an indeterminate form and not 1.
 
  • #15
DrewD said:
Everytime I have encountered it, ##0^0## has been considered an indeterminate form and not 1.

I agree, but let's not open that can of worms here. Paisiello2 clearly meant ##0^0=1##, so let's keep it at that, even if it's nonstandard.

mod note: not a ##0^0## argument please. All posts on this will be deleted.
 
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Related to Function that equals 1 at x=0, but 0 everywhere else?

1. What is a function that equals 1 at x=0, but 0 everywhere else?

A function that equals 1 at x=0, but 0 everywhere else is commonly known as the Dirac delta function. It is a mathematical function that is defined as 0 for all values of x except at x=0, where it has a value of infinity, and its integral from -∞ to +∞ is equal to 1.

2. What is the purpose of the Dirac delta function?

The Dirac delta function is commonly used in mathematics and physics to model point sources or point particles. It is also used as a mathematical tool to simplify calculations and solve differential equations.

3. How is the Dirac delta function represented mathematically?

The Dirac delta function is represented mathematically as δ(x), with the symbol "δ" resembling an arrow pointing to the point x=0. It is also sometimes written as δ(x-0) to indicate that it is a function of x with a value of 0 at all points except at x=0.

4. Can the Dirac delta function be graphically represented?

No, the Dirac delta function cannot be graphically represented as it is a point function with a value of infinity at x=0. However, it can be represented as a spike or a vertical line at x=0 on a graph with a very small width.

5. What are some real-world applications of the Dirac delta function?

The Dirac delta function has various applications in physics, engineering, and signal processing. It is used to model point particles in quantum mechanics, calculate impulse responses in electrical circuits, and analyze signals in digital signal processing. It is also used in the field of image processing to enhance and sharpen images.

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