Suppose that f is uniformly continuous on a bounded half-open interval (a,b]. Show that lim_x-> a+ f(x) exists and is finite. Is the same conclusion true for every bounded continuous function on (a,b]? (If yes show a proof, if no show a counter example).
For this should I use the cauchy criterion theorem which says suppose x_0 in R and f is defined for all x is some deleted neighborhood of x_0 . Suppose for all epsilon >0 there exist delta>0 such that if x,y in N and if 0<|x-x_0|<delta and 0<|y-x_0|<delta then |f(x)-f(y)|<epsilon. then lim_x->x_0 f(x) exists and is finite.
How would i actually use that?