let f: I -> R, where I is an open interval and f is a monotone function. Assume f is discontinuous at c. show that lim x→c- f(x) and lim x→c+ f(x) exist and are finite
The Attempt at a Solution
Well since f is discontinuous at c we know there exists a ε0, for all δ > 0 such that |x-c| < δ and |f(x)-f(c)| ≥ ε0.
So from that we know,
f(c) - ε0 ≥ f(x) ≥ f(c) + ε0
so it would seem that f(x) is bounded. If it is bounded and, WLOG since it is monotone, increasing it will converge to the sup. And i suppose that sup is the limit? This is where i get confused and where i get lost in how to show all of this using symbols and what not