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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?
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.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?
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.
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.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.
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 understand, but I don't think it's necessary to qualify the term "continuous" at an endpoint of the domain.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.
At an endpoint of the domain, the approach to c necessarily can only be one-sided.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).
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.
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.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.
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.
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?
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.
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##?
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?
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?
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.
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)##.
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##
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.
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.
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.
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.
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.