Function

There is a function satisfying the criterion, in fact infinitely many.

BTW, the function f(x) = sin x is a counterexample to your conjecture isnt it?

Chinmay's post offers a good starting point

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check if the function constructed by you is continuous.

You can easily construct more examples.

Take: f(x) = x if -1≤x≤1,
= 2 -x if 1 ≤x≤3
f(x+4) = f(x)

And now consider g(x) = e^|x| f(x)

yup sorry

f(x) = lim_{a→∞}(ax sin x) works..

same concept as e^|x| sinx

is der any other type of function? i.e w/o sine?

To add to aditya's wonderful example we look at how many solutions we have for

mx+c = e^{|x|} \sin x

or e^{|x|} \sin x - mx = c

For large enough n, if we consider the points

x = (4n+1)\frac{\pi}{2}; (4n-1)\frac{\pi}{2} the function oscillates from a negative number of large magnitude to a large +ve number.

By IMV it has to take the value c at some point between these two.

Further we will obtain solutions for all N≥n so that now we have infinitely many intersections with the line y = mx+c

[@ Arnab it means all non-vertical lines. Otherwise chinmay's example was good enough.]

@arnab- you are mistaken. the problem is to find a function such that it intersects ANY straight line at infinite points (non vertical). i.e. y=mx+c where m,c are parameters should intersect at infinite points

Completely misunderstood , thought any = all

Here is a simpler function

f(x)=x+ sin x

and its easy to see/prove that it intersects y=x at infinte points.

edit: log in problem :D

good work! but the onus is now on you to prove it :D

Ofcourse!

take f(x)=sin x and the non-vertical line as y=some legitimate k like 0.5(say) and you are through!

Guys, please attempt to substantiate your claims. That itself will weed out several guesses

hmmm..may be

this is the graph of y = 6897 sin x

what about y=Asinx where A→∞ ?
any non-vertical line will be cut infinite times

let the line be: y=x

and the function be y=|x| for all x e (-∞,-1) u (1,∞)
=x² for all x e (-1,1)

if its like this
i think it is possible
but if its like this
it is not a function at all.

Hmm....I am able to think of a couple of functions continuous over their domain which intersect any non-vertical line at infinitely many points.
I doubt if it holds for any continuous function over entire real line...