Is only (i) true for a continuous function f with given conditions?

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In summary, the conversation discusses a question involving determining which statements are true. The speaker plots the line y=3 and three given points to draw conclusions. They then discuss two possible ideas for the function f and come to the conclusion that only the first statement must be true.
  • #1
Umar
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View attachment 6010

Hi there, I'm having trouble with the above question. Basically, I need to determine which, or all of the statements are true. I've tried coming up with different ways the function can look like to satisfy or not satisfy the statements, but have come to no luck in doing so. If anyone could assist me on this one, that would be appreciated.
 

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  • #2
I would begin by plotting the line $y=3$ and the 3 given points:

$(1,3),\,(4,5),\,(5,3)$

What conclusions do you now draw regarding the 3 statements?
 
  • #3
MarkFL said:
I would begin by plotting the line $y=3$ and the 3 given points:

$(1,3),\,(4,5),\,(5,3)$

What conclusions do you now draw regarding the 3 statements?

View attachment 6011

This is a rough sketch of two possible ideas I had the function f might look like, the blue line being one possibility where f(0)>3 and f(2) > 3.

The purple line only satisfies f(2) > 3 and f(6) < 3.
 

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  • #4
Okay, good! (Yes)

Which, if any, of the given conditions must be true then?
 
  • #5
MarkFL said:
Okay, good! (Yes)

Which, if any, of the given conditions must be true then?

Hmm, the only think I can think of which has to be true is the first condition... since the other ones can be changed
 
  • #6
Umar said:
Hmm, the only think I can think of which has to be true is the first condition... since the other ones can be changed

I agree...only (i) must be true. With $f(1)=3$ and $f(4)=5$ and no other possibilities for $f(x)=3$ on $(1,4)$, then we must have $f(2)>3$. You demonstrated the other two don't have to be. :D
 
  • #7
MarkFL said:
I agree...only (i) must be true. With $f(1)=3$ and $f(4)=5$ and no other possibilities for $f(x)=3$ on $(1,4)$, then we must have $f(2)>3$. You demonstrated the other two don't have to be. :D

Thank you so much!
 

FAQ: Is only (i) true for a continuous function f with given conditions?

1. What does it mean for a function to be continuous?

Continuity of a function means that the function is defined at every point in its domain and there are no abrupt changes or gaps in its graph.

2. How do you prove that a function is continuous?

To prove that a function is continuous, you must show that it satisfies the three conditions of continuity: it is defined at every point in its domain, the limit as x approaches a of f(x) exists, and the limit as x approaches a of f(x) is equal to f(a).

3. What is the importance of continuity in mathematics and science?

Continuity is important in mathematics and science because it allows us to make predictions and draw conclusions based on the behavior of a function at specific points. It also allows us to use techniques such as differentiation and integration to solve problems.

4. Can a function be continuous at some points and discontinuous at others?

Yes, a function can be continuous at some points and discontinuous at others. This is known as a piecewise continuous function, where the function is defined differently on different intervals of its domain.

5. How does continuity relate to differentiability?

Continuity is a necessary condition for differentiability. If a function is not continuous at a point, it cannot be differentiable at that point. This means that for a function to be differentiable, it must also be continuous at every point in its domain.

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