Sage-Tips

Do there exist onto continuous function fଵ 0 1 → Q?

Originally Answered: What is the function f:[0,1] -> R such that f is continuous on Q and discontinuous on [0,1] \Q? There is no function which is continuous only at the rationals. The key point is that the set of discontinuities is an set, i.e. it is a countable union of closed sets.

What does the intermediate value theorem guarantee?

The word value refers to “y” values. So the Intermediate Value Theorem is a theorem that will be dealing with all of the y-values between two known y-values. In other words, it is guaranteed that there will be x-values that will produce the y-values between the other two if the function is continuous.

Why do we need continuity for the intermediate value theorem?

The Intermediate Value Theorem guarantees that if a function is continuous over a closed interval, then the function takes on every value between the values at its endpoints.

Does the Intermediate Value Theorem guarantee that there is a real zero between 3 and 4?

We see that one zero occurs at x = 2 \displaystyle x=2 x=2. Also, since f ( 3 ) \displaystyle f\left(3\right) f(3) is negative and f ( 4 ) \displaystyle f\left(4\right) f(4) is positive, by the Intermediate Value Theorem, there must be at least one real zero between 3 and 4.

Why is the Intermediate Value Theorem important?

this theorem is important in physics where you need to construct functions using results of equations that we know only how to approximate the answer, and not the exact value, a simple example is 2 bodies collide in R2. in this case you will have system of 2 equations in similar form to the example of the first part.

What is the intermediate value theorem in calculus?

Intermediate Value Theorem Statement Intermediate value theorem states that if “f” be a continuous function over a closed interval [a, b] with its domain having values f (a) and f (b) at the endpoints of the interval, then the function takes any value between the values f (a) and f (b) at a point inside the interval.

How do you prove that f(x) = c?

Use the Intermediate Value Theorem to prove f: [ 0, 1] → [ 0, 1] continuous and C ∈ [ 0, 1], there is some c ∈ [ 0, 1] such that f ( c) = C. Using a similar technique to the proof of the intermediate value theorem, I can easily prove that there is an f ( x) = C, but I am having trouble proving that a f ( c) = C.

How do you find the interval value of a function?

INTERMEDIATE VALUE THEOREM: Let $f$ be a continuous function on the closed interval $ [a, b] $. Assume that $m$ is a number ($y$-value) between $f (a)$ and $f (b)$. Then there is at least one number $c$ ($x$-value) in the interval $ [a, b]$ which satifies $$ f (c)=m $$

How do you explain the K-theorem?

This theorem is explained in two different ways: If k is a value between f (a) and f (b), i.e. then there exists at least a number c within a to b i.e. c ∈ (a, b) in such a way that f (c) = k