Not exactly. The terms bounded, open, closed, maximum, minimum, supremum, infimum are actually from set theory and can be applied to any set of numbers. An interval is a particular type of set of numbers. It is merely a chunk of one axis. A function can certainly be bounded or unbounded. Tanx is unbounded but sinx is bounded.It is the interval that bounds not the function itself bounds
In order to help clarify I am attaching an extract from a really good and easy to understand book that I recommend. He uses curved brackets for open intervals rather than reversed ones.It is an important theorem that a set which is bounded above has a greatest member and a set which is bounded below has a least member.
Exactly.But in this statement in Numerical Analysis,it is the domain interval that bounds not the range of the function bounds.Not exactly. The terms bounded, open, closed, maximum, minimum, supremum, infimum are actually from set theory and can be applied to any set of numbers. An interval is a particular type of set of numbers. It is merely a chunk of one axis. A function can certainly be bounded or unbounded. Tanx is unbounded but sinx is bounded.
Ashamed.I have not recognize it,but it should be true to bounded intervals(or say,functions that is contiunous on bounded range).It is an important theorem that a set which is bounded above has a greatest member and a set which is bounded below has a least member.
As PapaBravo says,closed intervals aren't bounded intervals.So the statement and the textbook are not lossing terminology.I think the book has slightly loose terminology.
An interval may be closed in which case it includes its endpoints
or open in which case it does not.
The orignal statement shows that the interval is for domain."Given any function,defined and continuous on a closed and bounded interval,there exisits a polynomial that is as "close" to the given function as desired.".
Good question, you were quick to spot the discrepancy.According to the papers you gave,intervals [a,b],[a,+∞) and (-∞,b] are closed.So interval of (-∞,0] is closed.(In the ISO notation,does ]-∞,0] equal to [-∞,0]?)
(-∞, a) and (a, +∞) are open and not closedGood question, you were quick to spot the discrepancy.
Note that Binmore avoids writing [-∞ anywhere. He only puts open brackets (-∞ around infinities.
I did say that people get into difficulties when employing ∞. I have not seen any author use the closed bracket around infinities.
The difficulty is that the interval (-∞, +∞) or ]-∞, +∞[ is regarded as both open and closed, but never written [-∞, +∞]
The intervals (-∞, a) and (a, +∞) are generally regarded as closed (as Binmore says) but written as if they were open at the infinity end.
The invtervals (-∞,a) and (a,+∞) are open and not closed,they have not got their all accumulation/limit point,say a.The intervals (-∞, a) and (a, +∞) are generally regarded as closed (as Binmore says) but written as if they were open at the infinity end.
by Jake Hertz
by Jake Hertz
by Duane Benson