Limit of a function can be found from the graph of that function, besides other methods. Some of the graphs are continuous. So what is continuity? That means they can be drawn without lifting pencil from the paper. See some examples below:
The functions that all the above graphs represent are continuous. Now look at the following graphs:
All the above graphs are not continuous or discontinuous as they cannot be drawn without lifting the pencil from the paper. With this understanding now let us try to define continuity.
Definition of continuity:
If the domain of a real function f contains an interval containing a and if lim (x->a) f(x) exists and lim (x->a) f(x) = f(a), then we say that f is continuous at x = a.
Thus, if lim (x->a+) f(x), lim (x->a-) f(x) and f(a) all exist and are equal, the f is said to be continuous at x = a.
If f is not continuous at x = a, we say that it is discontinuous at x = a.
(1) In the following picture, x is not defined at x = 2. Therefore f is discontinuous at x = 2.
(3) The above picture, at x = 1, both left and right hand limits exist and are equal but the limit of the function is not equal to f(1) so the function is again discontinuous at the point x = 1.
In simple words we can state continuity as follows:
A function is said to be continuous at any point x = a if the following three conditions are met:
(a) f(a) exists
(b) lim (x->a-) f(x) = lim (x->a+) f(x) = lim(x->a) f(x)
(c) quantities in (a) and (b) are equal.
If any of the above conditions is not met, we say that the function is discontinuous at the point x = a.
