Definition of a Continuous function

Is the epsolon-delta definition of a continuous function stating that however small a neighbourhood around 'f(c)' we want f(x) to be, we can choose a sigma such that if the distance between x and c is less than sigma, the former is implied. Ok not very well expressed but you know what I mean. Then I recognised that this is simply the definition of a limit when L=f(c). So what is the definition reffering to. The limit, or the the intuitive notion that small changes in x produce small changes in f(x). I feel there is a cause and effect idea I'm missing. Thanks