Imvt & LMVT: Existence of Point with f(x)=x

  • Thread starter Thread starter Ananya0107
  • Start date Start date
AI Thread Summary
A continuous function f:[0,1]→[0,1] must intersect the line y=x, indicating that there exists a point where f(x)=x. This is a standard application of the Intermediate Value Theorem (IVT) and Mean Value Theorem (MVT). The argument presented involves constructing a unit square and demonstrating that the graph of f must intersect the diagonal of the square. By applying MVT, it is shown that there exists a point c where the derivative f'(c) equals 1, supporting the existence of such a point. The discussion emphasizes the necessity of continuity in proving the existence of fixed points in this context.
Ananya0107
Messages
20
Reaction score
2
Let f:[0,1]→[0,1] be a continuous function . Show that there exists a point such that f(x) = x
 
Mathematics news on Phys.org
That's a fairly standard homework exercise. What have you tried?
 
Tell me if this argument is right... I constructed a 1by 1 square on the coordinate plane vertices (0,0) ,(0,1),(1,0),(1,1). Then I said that to graph any continuous function , the graph must touch or cut the square's diagonal , that is y=x.
 
Apply MVT.
According to MVT, there exists a point c such that, f'(c)={f(1)-f(0)}/(1-0)=1-0=1
Now dy/dx=1. So you can find y.
now substitute a point in the domain to get the value of integration constant.
 

Thread 'Non-orthogonal bases'

Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
View full post »

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Similar threads

MHB Solve Image of f(x)=-2(x+1)/(x-1)^2 in x from [0,1)U(1,3]
Replies
2
Views
1K
MHB What is the Range of the Function f(x) = (1+x)^0.6 / (1+x^0.6) for x in [0,1]?
Replies
2
Views
1K
I My attempt to understand horizontal transformations of functions
Replies
6
Views
1K
A Why the triangle inequality is greater than the 2 max{f(x),g(x)}
Replies
1
Views
1K
I Is g(x) Equal to g(a) If Their Integrals Are Equivalent?
Replies
20
Views
2K
MHB Is $f(x) = xf(1)$ the only solution to the given functional equation?
Replies
3
Views
1K
MHB Find max(abs(a)+abs(b)+abs(c)), and a possible f(x)
Replies
3
Views
1K
MHB Proving Equivalence of f(x) and g(x)
Replies
5
Views
1K
B Can We Simplify the Taylor Series Expansion for e^(f(x,y))?
Replies
3
Views
2K
B Inverse Functions: Why rewrite as y=f(x) ?
Replies
11
Views
2K

Hot Threads

Recent Insights

Back
Top