Continuous Function: Exploring Definition & Difference

In summary, the domain of a function is fundamentally part of the function's definition and when we say a function is continuous on its domain, we mean that it is continuous at each point in its domain. This is different from simply saying a function is continuous, which implies that it is continuous at all points where it is defined. A function can be continuous on its domain but not at certain endpoint points, as the approach to these points can only be one-sided. The epsilon-delta definition of continuity holds for all points in the domain, including endpoints, and only considers points within the domain of the function. Therefore, the domain is an essential aspect in determining the continuity of a function.
  • #1
Gurasees
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What do we mean when we say a function is continuous on its domain? How is that different from simply saying that a function is continuous?
 
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  • #2
Gurasees said:
What do we mean when we say a function is continuous on its domain? How is that different from simply saying that a function is continuous?
A function is always associated with its domain, so if someone says a function is continuous, the implication is that it is continuous at each point in its domain.
 
  • #3
Gurasees said:
What do we mean when we say a function is continuous on its domain? How is that different from simply saying that a function is continuous?
Mark44 said:
A function is always associated with its domain, so if someone says a function is continuous, the implication is that it is continuous at each point in its domain.

Yes, but a continuous function f: [a,b] ⊂ R→ R is in the following way defined:

f is continuous in (a,b)
f is right continuous in a
f is left continuous in b

So, we cannot truly say that a continuous function is continuous in every point of its domain, as this function is not continuous in a and b.
 
  • #4
Math_QED said:
Yes, but a continuous function f: [a,b] ⊂ R→ R is in the following way defined:

f is continuous in (a,b)
f is right continuous in a
f is left continuous in b

So, we cannot truly say that a continuous function is continuous in every point of its domain, as this function is not continuous in a and b.
I might be wrong, but when the domain of a function is a closed interval (such as f(x) = ##sqrt{x}##), we can say that f is continuous on its domain, and it is understood that we mean the continuity is one-sided at 0.
 
  • #5
Mark44 said:
I might be wrong, but when the domain of a function is a closed interval (such as f(x) = ##sqrt{x}##), we can say that f is continuous on its domain, and it is understood that we mean the continuity is one-sided at 0.

I agree, but you said that a continuous function is continuous on every point of its domain and obviously, the function
##f: [0,+∞) → ℝ: x → √{x}## is not continuous at x = 0, although it is a continuous function (see the definition above). It's one-sided continuous in x = 0 as you stated yourself. Of course, these are details, and I'm sure you are aware of this fact, but I just wanted to point that out.
 
  • #6
Math_QED said:
I agree, but you said that a continuous function is continuous on every point of its domain and obviously, the function
##f: [0,+∞) → ℝ: x → sqrt{x}## is not continuous at x = 0, although it is a continuous function (see the definition above). It's one-sided continuous as you stated yourself. Of course, these are details, but I just wanted to point that out.
I understand, but I don't think it's necessary to qualify the term "continuous" at an endpoint of the domain.

From Wikpedia (https://en.wikipedia.org/wiki/Continuous_function), FWIW (emphasis added):
Definition in terms of limits of functions
The function f is continuous at some point c of its domain if the limit of f(x) as x approaches c through the domain of f exists and is equal to f(c).
At an endpoint of the domain, the approach to c necessarily can only be one-sided.
 
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  • #7
Mark44 said:
I understand, but I don't think it's necessary to qualify the term "continuous" at an endpoint of the domain.

From Wikpedia (https://en.wikipedia.org/wiki/Continuous_function), FWIW (emphasis added):

At an endpoint of the domain, the approach to c necessarily can only be one-sided.

Then we are in agreement.
 
  • #8
When I say that a function is continuous, it is understood that it is continuous in its '' natural '' domain. For example If I say that ##f(x)=\frac{1}{x}## is continuous, it is understood that its natural domain is ##\mathbb{R}\setminus \{0\}## ...
 
  • #9
Math_QED said:
I agree, but you said that a continuous function is continuous on every point of its domain and obviously, the function
##f: [0,+∞) → ℝ: x → √{x}## is not continuous at x = 0, although it is a continuous function (see the definition above). It's one-sided continuous in x = 0 as you stated yourself. Of course, these are details, and I'm sure you are aware of this fact, but I just wanted to point that out.
That's not right at all. When considering the domain as ##[0, +\infty)## the epsilon-delta definition of continuity holds completely as there are no points less than ##0## under consideration. Right and left sided limits don't come into it.
 
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  • #10
PeroK said:
That's not right at all. When considering the domain as ##[0, +\infty)## the epsilon-delta definition of continuity holds completely as there are no points less than ##0## under consideration. Right and left sided limits don't come into it.

I thought that a function is continuous in a point x = a if lim x-> a f(x) = f(a). I haven't seen the epsilon-delta definition yet, but for the function f(x) = √x: lim x-> 0 f(x) does not exist as we can't approach f(x) from the left? Or did I miss something obvious?

EDIT: I read this: http://math.stackexchange.com/questions/637280/limit-of-sqrt-x-as-x-approaches-0 . This cleared a lot up for me. Thanks for pointing that out @PeroK.
 
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  • #11
Consider f(x) = x defined for integer x. Is f continuous?

Under the epsilon-delta definition, this function is continuous at every point in its domain.
 
  • #12
Math_QED said:
I thought that a function is continuous in a point x = a if lim x-> a f(x) = f(a). I haven't seen the epsilon-delta definition yet, but for the function f(x) = √x: lim x-> 0 f(x) does not exist as we can't approach f(x) from the left? Or did I miss something obvious?

More generally, when you consider a function defined on a more general set, you no longer have the temptation to consider points not in its domain.

Because you are so familiar with the real line, you were tempted to consider the function's behaviour for negative x.

If you think of a function defined on, say, the set of invertible matrices, then why would you consider its behaviour outside of this set?

In summary, the domain of a function is fundamentally part of the function's definition. It's a good lesson in pure mathematical thinking to be able to focus on the domain and not see a different function defined on a larger domain.
 
  • #13
PeroK said:
That's not right at all. When considering the domain as ##[0, +\infty)## the epsilon-delta definition of continuity holds completely as there are no points less than ##0## under consideration.

You might be correct, but we could subject that to technical analysis. Is there any limitation on x in the condition "## |x - a| < \delta##"? Is the statement " ## | f(x) - L | < \epsilon ## " true when ##x## is not in the domain of ##f##?

Things might be clearer if we use the topological definition of continuity in terms of open sets. But then we have the question of which toplogy to use. For real valued functions, do we use the standard topology of the real line or do we use the relative topology it induces on the domain of the function?

If we really want to drive ourselves crazy, we can pose a similar question about uniform continuity.
 
  • #14
Stephen Tashi said:
You might be correct, but we could subject that to technical analysis. Is there any limitation on x in the condition "## |x - a| < \delta##"? Is the statement " ## | f(x) - L | < \epsilon ## " true when ##x## is not in the domain of ##f##?

The epsilon delta is very clear: you only test points in your domain.
Edit: realized that the link is to limits not continuity. Strike that.

Things might be clearer if we use the topological definition of continuity in terms of open sets. But then we have the question of which toplogy to use. For real valued functions, do we use the standard topology of the real line or do we use the relative topology it induces on the domain of the function?

The former gives ##[0,\infty)## as neither open nor closed, so its not a topology for ##[0,\infty)##.
 
  • #15
Stephen Tashi said:
Things might be clearer if we use the topological definition of continuity in terms of open sets. But then we have the question of which toplogy to use. For real valued functions, do we use the standard topology of the real line or do we use the relative topology it induces on the domain of the function?

If a function is given then the topology for the domain and codomain must also be given. Then continuity is defined in terms of preimages of open sets being open as well.

Now, regardless of which topology ##[a,b]## is equipped with, continuity makes sense at the end points. A function ##f: [a,b] \rightarrow \mathbb{R}## is continuous at ##a## if for each open set containing ##f(a)##, there is an open set ##U \ni a## such that ##U \subset f^{-1}\big(f(a)\big)##. A similar definition is given for continuity at ##b##. You'll notice that this definition actually makes sense for any point ##x \in [a,b]##. So we can talk about continuity at the end points of a closed interval and in fact it is not, in any sense, a sort of "special" continuity. It's the same old definition.
 
  • #16
pwsnafu said:
The epsilon delta is very clear: you only test points in your domain.
Edit: realized that the link is to limits not continuity. Strike that.

It's very clear in the Wikipedia, but a point of interest is whether that "precise definition" in the Wikipedia is equivalent to the epsilon-delta definitions in typical calculus books. Perhaps modern day books are more precise than books of a decade ago. If we speak of "the" epsilon-delta definition of limit then what will a calculus student understand that to mean if he consults his book? Answering the original post is going to involve emphasizing that there is a precise epsilon-delta definition as well as some not-so-precise epsilon delta definitions.

JonnyG said:
A function ##f: [a,b] \rightarrow \mathbb{R}## is continuous at ##a## if for each open set containing ##f(a)##, there is an open set ##U \ni a## such that ##U \subset f^{-1}\big(f(a)\big)##.

Somewhere in that definition, you need a symbol representing the "open set containing ##f(a)##".
 
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  • #17
Yes, I meant to write that there is a neighborhood ##U \ni a## such that ##f(U) \subset \mathrm{the open set containing } f(a)##

I will edit my original post to reflect this.

EDIT: I can't edit my original post anymore. So the topological definition of continuity at a point: ##f: X \rightarrow Y## is continuous at a point ##a \in X## if for every open set ##V \ni f(a)##, there is an open set ##U \ni a## such that ##f(U) \subset V##
 
  • #18
JonnyG said:
So the topological definition of continuity at a point: ##f: X \rightarrow Y## is continuous at a point ##a \in X## if for every open set ##V \ni f(a)##, there is an open set ##U \ni a## such that ##f(U) \subset V##

And it would simplify the life of students of analysis if they learned this definition early in their studies, instead of approaching ##\mathbb{R^2}, \mathbb{R^3} ## as completely new vistas. One can make a nice analogy to the epsilon-delta definition by using "E" to denote the open set containing ##f(a)## and "D" to denote the open set containing ##a##.
 
  • #19
Hey Gurasees.

The intuitive idea of continuity is that if you were able to draw a continuous mapping (like a function - but not necessarily) then you could do it by using a marking element (like a pencil) and you would be able to represent the information without taking the marking element off of the paper.

It means that if you can do that then the object you are representing is continuous - if you have to take the marking element off for some reason then it isn't.

That is basic continuity and formally it means saying that any limits from the left and right equal the actual value at that point across all of the points you consider.

If for some reason you need to take the marking element off then it violate the limit conditions and that shows the discontinuity part.
 

Related to Continuous Function: Exploring Definition & Difference

What is a continuous function?

A continuous function is a type of mathematical function in which the outputs change smoothly as the inputs change. This means that there are no abrupt jumps or breaks in the graph of the function, and the function can be drawn without lifting the pencil from the paper.

What is the difference between a continuous function and a discontinuous function?

The main difference between a continuous function and a discontinuous function is that a continuous function has no breaks or interruptions in its graph, while a discontinuous function has at least one point where the graph is interrupted or has a jump. Another way to think about it is that a continuous function can be drawn without lifting the pencil from the paper, while a discontinuous function cannot.

How is continuity of a function determined?

The continuity of a function is determined by three main criteria: the function must be defined at the point in question, the limit of the function at that point must exist, and the value of the function at the point must be equal to the limit. If all three criteria are met, then the function is considered continuous at that point.

What are some examples of continuous functions?

Some examples of continuous functions are linear functions, polynomial functions, exponential functions, and trigonometric functions such as sine and cosine. These functions can be drawn without lifting the pencil from the paper and have no breaks or interruptions in their graph.

Why is the concept of continuity important in mathematics and science?

The concept of continuity is important in mathematics and science because it allows us to model and understand real-world phenomena more accurately. Many natural processes and physical systems can be described using continuous functions, and the concept of continuity allows us to make predictions and analyze these systems more effectively. Additionally, many advanced mathematical concepts, such as derivatives and integrals, rely on the concept of continuity.

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