This is somewhat confusing A function f : (a,b) → R is uniformly continuous if and only if the limits L a := lim_(x→a) f(x) and L b := lim_(x→b) f(x) exist and the function ˜f : [a,b] → R defined by...

This is somewhat confusing

A function f : (a,b) → R is uniformly continuous if and only if the limits
L_a
:= lim_(x→a) f(x) and L_b
:= lim_(x→b) f(x) exist and the function ˜f : [a,b] → R defined by

˜f(x) := {f(x) if x ∈ (a,b), and L_a
if x = a, and L_b
if x = b, is continuous.

Let f : (a,b) → R be a uniformly continuous function. Prove that the limit lim_(x→b) f(x) exists.