# Struggling with a problem in functions.

Suppose '$f$' is a continuous function from $\mathbb{R}$ to $\mathbb{R}$ and $f(f(a))=a$ for some $a \in \mathbb{R}$ then find the number of solutions of the equation $f(x)=x$.

Options given:

(a) no solution

(b) exactly one solution

(c) at most one solution

(d) atleast three solutions

I tried to solve it like this

$$f(f(a))=a$$ $$f(a)=f^{-1}(a)$$

So the value of function is equal to its inverse at $x=a$.

I tried thinking about that function but it will become a special case.

Is there a mathematically rigorous and simple way to tackle this problem?

• Is $f$ given in your situation? – wythagoras Jun 7 '15 at 11:42
• No, it's not given. @Wythagoras – me_ravi_ Jun 7 '15 at 11:44

You can say there is at least one solution to $$f(x)=x$$. There may be more, but there is at least one.

Here is a proof. If $$f(a)=a$$ then $$a$$ is a solution and we are done. Assume for now that $$f(a).

Consider the function $$g(x)=f(x)-x$$. Note that $$g$$ is continuous since $$f$$ is continuous. Then

$$g(a)=f(a)-a<0$$

and

$$g(f(a))=f(f(a))-f(a)=a-f(a)>0$$

Hence, by the Intermediate Value Theorem for continuous functions, $$g(x)$$ has a zero between $$f(a)$$ and $$a$$. Let's call that zero $$c$$. Then

$$0=g(c)=f(c)-c$$

so $$f(c)=c$$ and we have found a solution.

You can repeat this argument if $$f(a)>a$$. That covers all possibilities regarding $$f(a)$$ and $$a$$, so there is at least one solution to $$f(x)=x$$.

In the edited version of your question, which gives four multiple choice possible answers, none of those choices is correct. An example where there is exactly one solution is $$f(x)=-x$$, which disproves choices (a) and (d). An example where there are infinitely many solutions is $$f(x)=x$$. This disproves choices (b) and (c).

Are you sure there was no choice "at least one solution"?

• The answer given in my book is (d) atleast three solutions. @RoryDaulton – me_ravi_ Jun 7 '15 at 11:57
• @Integrator: Then your book is wrong, as the example $f(x)=-x$ shows. – Rory Daulton Jun 7 '15 at 11:59
• Thanks @RoryDaulton I think the author overlooked the fact that given question ( with given options ) does not suffice for $f(x)=-x$. This example is great. – me_ravi_ Jun 7 '15 at 12:42