Explain why the function is discontinuous at the given point a. For the function f(x) = 1 / ( x - 3 ), find the point of discontinuity.

We will use the concept of limits, continuity, and differentiability to find out why a function is discontinuous at a given point a.

Answer: The function f(x) = 1 / (x - 3) is discontinuous at x = 3. The explanation for the discontinuity of a function at a point is given below.

Let us see how we will use the concept of limits, continuity, and differentiability to find out why a function is discontinuous at a given point a.

Explanation:

There can be several reasons that why a function becomes discontinuous at a given point a.

1) The given point is not in the domain of the function. For example : ln (x) is discontinuous at x = 0 , because x = 0 is not in the domain of ln (x).

2 ) Left-hand limit of the curve is not equal to the value of the function at that point. For example: sin | x | / x is discontinuous at x = 0.

3 ) Right hand limit is not equal to the value of function at that point. For example : sin | x | / x is discontinuous at x = 0.

4) The value of the function at a is not equal to the limit of the function as x approaches to a.

For example, in the function f(x) = 1 / (x - 3), suppose we find the domain of the function then, the domain of function f(x) = 1 / (x - 3) is every real number except x = 3, because at x = 3 the function is undefined. 

Hence, at x = 3 the function f(x) = 1 / (x - 3) is discontinuous.