uniformly continuous versus continuous

Question

It's hard to understand the difference between uniformly continuous function and continuous function.

So if A is a uniformly continuous function on X

and if B is a continuous function on X,

the only difference is that we can find one common delta value >0 that applied to epilson-delta conditions for all x values on X in A

while there is no such common delta value in B?

Answer 1

yes. The $\delta$ for continuous functions is dependent on $x$, as in the definition of continuity. On the other hand, the common $\delta$ for uniform continuity doesn't depend on $x$. That is, the $\delta$ works for all $x$'s in the domain.

Answer 2

Consider the following example : $X=[0,1)$, $A(x)=x$, $B(x)=1/(1-x)$

Given $\epsilon$, there exists $\delta >0$ such that $$ |x-y| <\delta \Rightarrow | A(x)-A(y)|<\epsilon$$

But we cannot take such $\delta$ for $B$. Intuitively, two functions are increasing. But $A$ increases with same rate, i.e., its derivative is bounded, and $B$ increases " more rapidly " as $x$ goes to 1, i.e., its derivative goes to $\infty$.