MATLAB, 41 bytes

function x=g(f,x);while f(x)-x;x=f(x);end

There is also this beauty that does not need function files. Unfortunately it is a little bit longer:

g=@(g)@(f,x){@()x,@()g(g)(f,y)}{2-(y=f(x)==x)}();g(g)

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MATLAB, 41 bytes

function x=g(f,x);while f(x)-x;x=f(x);end

There is also this beauty that does not need function files. Unfortunately it is a little bit longer:

i=@(p,c)c{2-p}();g=@(g,f,x)i(f(x)==x,{@()x,@()g(g,f,f(x))});q=@(f,x)g(g,f,x)

Try it online!

MATLAB, 41 bytes

function x=g(f,x);while f(x)-x;x=f(x);end

Try it online!

MATLAB, 41 bytes

function x=g(f,x);while f(x)-x;x=f(x);end

There is also this beauty that does not need function files. Unfortunately it is a little bit longer:

g=@(g)@(f,x){@()x,@()g(g)(f,y)}{2-(y=f(x)==x)}();g(g)

Try it online!

Octave, 41 bytes

function x=g(f,x);while f(x)-x;x=f(x);end

Try it online!

MATLAB, 41 bytes

function x=g(f,x);while f(x)-x;x=f(x);end

Try it online!