>>13905631Close, but not quite. A function can be continuous everywhere without being uniformly continuous. Uniform continuity tells you even more. It tells you how consistent your level of continuity actually is, because the difference between values f(x) and f(y) is bounded by epsilon as long as the distance between values x and y is bounded by delta.
This is not the case for all continuous functions. For example, f(x)=x^2 is continuous everywhere, but it is NOT uniformly continuous because for any value e>0 and any value d>0, there exist x,y such that |x-y|<d, but not |f(x)-f(y)|<e. In particular, if we note that the derivative of f(x) is f'(x)=2x, then it can be seen that |f(x)-f(y)|>e whenever y=x+d and 2xd>e.
>>13905568A continuous function is one where we might have to work a little harder to force f(x) and f(y) to be close together. If f(x) is uniformly continuous, then our job easy, no matter which value of x we select, because we always know that the difference |f(x)-f(y)|<e whenever |x-y|<d., regardless of the actual values of x and y If the function is not uniformly continuous, however, our job is slightly harder, since we might have to select smaller values of d>0 for certain points x in the domain.
