Prove that if is continuous and for all , then there is a contstant , such that , for all .
Here is my attempt:
Let and for all , then there is a contstant , such that , for all . Then, assume that f is NOT continuous. Then, suppose some point is not a limit point for . Then, since , is defined at . Now,let be a limit point for . Then, for some .
Then, is the same as .
Now evaluate . We have that . Thus, works. However, we have shown that if is not a limit point at then is defined at and that if is a limit point of then as . This is the definition of a continuous function so we have a contradiction. Thus, is continuous.
Any contructive critisism is welcome.