Limits: Discontinuity, a function with ‘breaks’

In Calculus, a branch of mathematics we learn about limit of a function, we say that the limit of f(x) is L as x approaches ‘a’ and is written as lim(x->a) f(x)=L, provided we can make f(x) as close to L as we want for all x sufficiently close to a, from both sides, without actually letting x be a.

Discontinuity : A function is said to be continuous at x=a if lim(x->a)f(x) = f(a). In simple words a function is said to be continuous if the graph of the function has no breaks in it, that is, it is a continuous curve. Many functions, however, will have isolated points where they are not connected. Such type of points is called Discontinuity points of that function. Definition of Discontinuity can be given as, a function is discontinuous at ‘a’ if it is defined at ‘a’ but is not continuous at ‘a’. Discontinuity points can be classified into three types.

The function f(x) has a discontinuity of first kind at x=a if there exists left hand limit, lim(x->a-0)f(x) and right hand limit (x->a+0 f(x) and also these one-sided limits are finite.
Point discontinuities are also called the removable discontinuities. Sometimes we come across functions that are defined differently for a certain point. Let us consider the function f(x)=1 for x=3 and f(x) =x^2,  for all positive real numbers. We define the value of the function to be one at the point x=3, though the rest of the function is given by f(x) = x^2. When we graph this function, we can see that the function is continuous except for this tiny hole in the curve at x=1. It is discontinuous at a single point x=1, and this discontinuity is called the point discontinuity or the removable discontinuity. In general, point discontinuities occur when a function is defined specifically for an isolated value of x.

What is Jump Discontinuity
One might wonder what is a Jump Discontinuity. Let us now learn about a Jump Discontinuity, in this type of discontinuity the right hand limit is not equal to the left hand limit of a function f(x). Jump discontinuities are also called simple discontinuities. Let us consider a function, f(x) = x^2 for x less than equal to 1 and f(x) =6-x^2 for x greater than 1. The two pieces have different value at x=1, and the graph of these function f(x) seem to ‘jump’ from one branch to another and this jump makes the function discontinuous. We refer this discontinuity as a jump discontinuity