Uniform continuity subtlety

Question

I have studied the concept of uniform continuity of a function , and I have been doubting on the following: If the function is continuous on the interval [a,b], we know that for every point y in it, we can enclose the functions value on any interval (f(y)-E, f(y)+E) for any number E>0 as long as we look at x values that belong to the interval (y-D,y+D) for some valid number D > 0 , which in general will be a function of the particular point and E in question. Now, I don’t see why for functions that satisfy this fact, we cannot consider as a “global” D, the “minimum” (or at least a lower bound) from all the ones that are used for each specific point y in te interval [a,b]. It makes sense intuitively, but I suppose this is not always possible in some cases because given a E , the set of D’s needed for satisfying the continuity condition for each y in the interval [a,b] may not be bounded below. I am pretty sure this is why we cannot assert that continuity means uniform continuity but I would like to see an example where this set of D’s has no lower bounds. Please, if someone can help and illustrate with a simple example I would appreciate it.