Is non continuous function also not Bounded ?

In summary, the conversation is about whether a non-continuous function in a closed interval is also not bounded. The person asking the question believes that this is the case, but another person points out that this is not necessarily true and provides a counterexample to prove it.
  • #1
Lancelot1
28
0
Dear all,

I am trying to figure out if a non continuous function is also not bounded. I know that a continuous function in an interval, closed interval, is also bounded. Is a non continuous function in a closed interval not bounded ? I think not, it makes no sense. How do you prove it ?

Thank you !
 
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  • #2
You are making an error in logic. The opposite of "if A then B" is NOT "if not A then not B". It is "if not A then we don't know anything about B"!

For example the functions f(x)= x for \(\displaystyle 0\le x\le 1\), f(x)= x+ 1 for \(\displaystyle 1\le x\le 2\) and g(x)= 0 if x is rational, g(x)= 1 if x is irrational are discontinuous (g badly discontinuous) but are bounded functions.
 

FAQ: Is non continuous function also not Bounded ?

What is a non-continuous function?

A non-continuous function is a mathematical function that has at least one point where it is not defined or has a break in its graph. This means that the function cannot be drawn as a continuous line.

What is a bounded function?

A bounded function is a mathematical function that has a finite range or set of values. This means that the function does not have any values that go to infinity.

Is a non-continuous function always not bounded?

No, a non-continuous function can still be bounded. For example, the function f(x) = 1/x is non-continuous at x=0, but it is still bounded between -1 and 1.

Is a non-continuous function always unbounded?

No, a non-continuous function can still be bounded. For example, the function f(x) = sin(1/x) is non-continuous at x=0, but it is still bounded between -1 and 1.

How can I determine if a non-continuous function is bounded or not?

To determine if a non-continuous function is bounded or not, you can look at the range of values for which the function is defined. If the function has a finite range, then it is bounded. If the function has values that go to infinity, then it is unbounded.

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