ILL 1977 Problem 46

Let $f$ be a strictly increasing function defined on the set of real numbers. For $x$ real and $t$ positive, set$$g(x,t)=\frac{f(x+t)-f(x)}{f(x) - f(x - t)}.$$ Assume that the inequalities$$2^{-1} < g(x, t) < 2$$ hold for all positive t if $x = 0$ , and for all $t \leq |x|$ otherwise. Show that$$14^{-1} < g(x, t) < 14$$ for all real $x$ and positive $t.$