Is f(x) a closed interval if f is continuous and onto on a bounded interval?

In summary, the theorem states that if f is continuous onto a closed interval, then the set of f(x) is also closed. However, if f is continuous onto an open interval, the range can be closed, open or neither.
  • #1
rapple
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1.Is it true that if f is continuous onto function on a closed interval then f(x) must also be a closed interval. How about the other way around. f is continuous and onto on a open bounded interval and f(x) is a closed interval

Homework Equations


f:[0,1]-->(0,1)
f:(0,1)-->[0,1]


The Attempt at a Solution


There is a theorem that says that if f is continuous on a closed and bounded interval then set of f(x) is a closed and bounded interval.
 
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  • #2
True, that is a theorem. f continuous on a closed interval then the image of f is a closed interval. If f is continuous on an open interval then the range can be closed, open or neither. Can you find an example of each?
 
  • #3
Dick said:
True, that is a theorem. f continuous on a closed interval then the image of f is a closed interval. If f is continuous on an open interval then the range can be closed, open or neither. Can you find an example of each?

Does this mean that f:[0,1]-->(0,1) does not exist. If both are closed intervals there are several examples, including the trivial f(x)=x

I don't have an example for continuous onto f:(0,1)-->[0,1]
 
  • #4
Yes, there is no continuous function f mapping [0,1]->(0,1). You cited the theorem. For mapping (0,1) into [0,1], can't you think of a function where f(0)=1/2, f(1/3)=1, f(2/3)=0 and f(1)=1/2? Surely you can draw one.
 
  • #5
Dick said:
Yes, there is no continuous function f mapping [0,1]->(0,1). You cited the theorem. For mapping (0,1) into [0,1], can't you think of a function where f(0)=1/2, f(1/3)=1, f(2/3)=0 and f(1)=1/2? Surely you can draw one.

It has to be an onto function. I am not aware of one as you have described.
 
  • #6
Onto what? It's certainly onto [0,1].
 
  • #7
Yes. But I don't have an example as you have described. Sorry. I am always bad with examples.
 
  • #8
Can't you linearly interpolate between the values I gave? You could also fit it to a polynomial. You could scale a sine function. Any number of things you could do to get an explicit example. You did sketch one, right? That's all I care about.
 
  • #9
Dick said:
Can't you linearly interpolate between the values I gave? You could also fit it to a polynomial. You could scale a sine function. Any number of things you could do to get an explicit example. You did sketch one, right? That's all I care about.

sin(4x) works

Thx
 

Related to Is f(x) a closed interval if f is continuous and onto on a bounded interval?

What is a continuous function?

A continuous function is a mathematical concept that describes a function whose graph is a single unbroken curve with no gaps or jumps. In other words, a continuous function can be drawn without lifting the pen from the paper.

How is continuity related to a function being onto?

A function is onto if every element in the range has at least one corresponding element in the domain. Continuity is related to this concept because if a function is continuous, then every point in the range can be reached from the domain without any gaps or jumps, ensuring that the function is onto.

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

A continuous function is one that has a smooth and unbroken graph, while a discontinuous function has gaps or jumps in its graph. This means that a discontinuous function is not considered to be continuous and may not have all points in the range connected to the domain.

Can a function be continuous but not onto?

Yes, it is possible for a function to be continuous but not onto. This can occur when the function has a limited range, meaning that not all elements in the range have corresponding elements in the domain. In this case, the function would be continuous within its defined range, but not onto.

How can I determine if a function is continuous onto?

To determine if a function is continuous onto, you can use the intermediate value theorem. This theorem states that if a function is continuous on a closed interval, and takes on values of both positive and negative, then it must take on every value in between. You can also check for one-to-one correspondence between the domain and range to ensure that every element in the range has a corresponding element in the domain.

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