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Suppose $a<b$ and $f:[a,b] \to [a,b]$ be continous. Suppose that $x \neq y$ in $[a,b]$ with $f(x)=y$ and $f(y)=x$. Prove that $f$ has a fixed point in $(x,y)$.

So I was thinking of considering the function $g(x)=f(x)-x$, which we know is continuous. Then we also know that because $f(a) \geq a$ that $g(a)=f(a)-a \geq 0$. Similarly, because $f(b) \leq b$ then $g(b)=f(b)-b \leq 0$.

Can we just use the fact that because $g(x)$ is continuous, $0 \in [g(b),g(a)]$, the IVT says there exists $c \in [a,b]$ such that $g(c)=f(c)-c=0$ so $f(c)=c$? Then we know $c$ is a fixed point.

How do we show that $c$ is in $(x,y)$??

We know that $g(x)=f(x)-x=y-x \neq 0$ and $g(y)=f(y)-y=x-y \neq 0$ but we don't know that those are in $(a,b)$?

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You've essentially stated the argument. But, rather restrict $f$ to $[x,y]$ (where w.l.o.g $x<y$). Then, define $g:[x,y]\to\mathbb R$ by

$$g(t)=f(t)-t$$

for any $t\in [x,y]$. As $f$ is continuous on $[a,b]$ and $[x,y]\subseteq [a,b]$, $g$ is continuous on $[x,y]$. Also, you have $g(x)=f(x)-x=y-x>0$ and $g(y)=f(y)-y=x-y<0$ as $x<y$.

Thus, by the intermediate value theorem, there is a $s\in (x,y)$ such that $g(s)=0$, i.e. $f(s)=s$.

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Without loss of generality you can assume that $x < y$. Now consider $g(t) = f(t) - t$ not on the entire interval $[a, b]$ but only on $[x, y]$.

Then $ g(x) = y- x$ and $g(y) = x-y$ have opposite sign, so that you can apply the intermediate value theorem.

Note also that I have chosen a different variable name ($t$ instead of $x$) for defining $g$, in order to avoid confusion between that variable and the given (fixed) value $x$.

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