Analyzing the range of temperature readings, we should say that the approach of x to x 0 had no bearing on the approach of the corresponding t ( x ) values to t ( x 0 ). Thus the point x 0 stands out as singular point (“singular” means “special” or “unusual”). In function notation we are assuming that

But, beyond x 0 suppose readings of 250° F are again observed. Then suppose that at x 0 the temperature drops suddenly to near room temperature, say 75° F, as if an insulation were at x 0. Suppose, to fix ideas, the temperature recordings are observed to be the same, 250☏, as one moves along the wire L until a point x 0 is reached on L. To each point x on the hot wire L is assigned a temperature readings t(x) on thermometer, T. A thermometer T measures T temperatures along a given hot wire L. Both continuity and limit are primarily concepts defined at a point, but continuity acquires a global character in a pointwise way.įor motivation, consider the following physical situation. In order to formulate the concept of continuity in terms of limit we must focus our attention at a point. Loosely speaking, to possess the property of continuity will mean “to have a favourable limit”. The proper and effective way of attitude which allows us to put the concept to work is to correlate continuity with limit. However, we will realise later that for functions with interval domains, continuity means essentially that the graph may be traced without lifting the point of the pencil. Accordingly, a premature use of the graph to gain insight into the meaning of continuity is advised against as gravely misleading. Nevertheless, to identify continuity with “no breaks in a graph” or “a hanging together” has serious drawbacks, at least from the point of view of applying the concept to the analysis of functions. In fact, the name itself derives from the Latin continuere, “to hang together”. The idea of continuity of a function stems from the geometric notion of “no breaks in a graph”. For instance, we speak of the continuous expansion of a rod on heating, of the continuous growth of an organism, of a continuous flow, or a continuous variation of atmospheric temperature etc. It reflects mathematically the general trait of many phenomena observed by us in nature.

One of the chief features in the behaviour of functions is the property known as continuity.

Fixing continuity calculus