Constructing a continuous function with a given property.

[tt2348]

Homework Statement

Find f:I->I such that each x on I shows up an infinite amount of times on f(I), f continuous

Homework Equations

Lol , equations?

The Attempt at a Solution

The weierstrauss function, I want to expand on it's fractal property and say that it crosses a given point infinite amounts of times. Confident this is right, I just don't know how I can manipulate it to stick on I, would squaring the cosine components and dividing by 2 restrict into a proper domain?

 

[mighty2000]

Thank you for your post. Your approach using the Weierstrass function is a good start. However, it is not continuous on the entire interval I, so it may not be the best choice for this problem.

One possible solution to this problem is to use a piecewise defined function. For example, you could define f(x) = x for x in [0,1) and f(x) = x+1 for x in [1,2). This function is continuous on the interval [0,2), and each x in [0,2) appears an infinite number of times on f(I).

Another approach is to use a trigonometric function, such as f(x) = sin(2πx). This function is continuous on the entire interval I and each x in I appears an infinite number of times on f(I).

I hope this helps. Good luck with your problem!
Scientist