What about a constant function?
If f(x)=c , where c is any constant then we can't find any a for which f(x)=f(x+a)
So I'm trying to find a solution to this question. In part (a), I have shown that for a function which is continuous on satisfying then for any number there is some for which I did this by considering the function and assuming for any We can then show that either or by noting that, since is continuous we must have or for all So, in the latter case for example, we have
This clearly relies on the fact that with
Now for part (b), which has me totally stumped. For with for any find a function continuous on satisfying for which for any I just can't see how that could work. Imagine any function, and connect the points and with a straight line - surely if we have this line must be horizontal at some point!
How do I find such an ?
Ok, I found the solution online. The graph of can be a saw function starting at and with which gradually climbs. Where it has local minima of at for and local maxima of at . Now you can just connect the minima and maxima with straight lines. If you sketch it, you'll see that for all appropriate and that .
Now I've seen a solution, I think this was probably easier than I was making out. Did get me stuck for a bit though
I found this to be an interesting challenging problem. Anyway, I thought up a solution similar to the one you describe, but thought I'd post it anyway. The first image is of such a function f (a is .3) and the dotted red graph is f(x+a):
The entire description is in two image files (I find it much easier to use my own math editor and paste images):
Thank you for that johng,
I'm really impressed (and a bit jealous!) that you achieved this on your own! Apologies for not putting a spoiler alert on the thread, I didn't realise people were working on it. My own image is drawn below with a bit of labelling in case anyone is confused by my sloppy description above!
The function from my previous post.
In case you're interested, the problem is question 19.(b) of chapter 7 from Calculus by Spivak. This book has lots of problems that make you think like this and I highly recommend it.