- #1

- 81

- 1

## Homework Statement

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

## Homework Equations

n/a

## The Attempt at a Solution

Well since f is discontinuous at c we know there exists a ε

_{0}, for all δ > 0 such that |x-c| < δ and |f(x)-f(c)| ≥ ε

_{0}.

So from that we know,

f(c) - ε

_{0}≥ f(x) ≥ f(c) + ε

_{0}

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