Unit – IA

Continuity of a Function at a Point

A Concept of Left and Right Limit of a Function:

  • Let ‘f’ be a real function defined on domain [1, 3] and the following limit exists.

  • Here, value of x can be such that 1  ≤ x ≤ 3
  • Now x can approach 2 from left-hand side by taking values like (1.9, 1.99, 1.999, …..). This left side approach is represented as

  • Similarly,  x can approach 2 from right-hand side by taking values (2.1, 2.01, 2.001, …..). This right side approach can be represented as

  • A Limit of the function exists if and only if the following condition is satisfied.

Concept of Continuity of a Function:

  • Let ‘f’ be a real function on a subset of the real numbers and let c be a point in the domain of ‘f’. Then ‘f’ is continuous at c if

  • Thus, if the left-hand limit, right-hand limit and the value of the function at x = c exist and are equal to each other, i.e.,
  • then ‘f’ is said to be continuous at x = c.
    Conditions for a function to be continuous at a point:
    1. Both left-hand limit and right-hand limit of function exist
    2. f(c) exists and
  • A graph for Continuous and Non-Continuous Function:

  • f(x) = sinx, is a continuous function

  • f(x) = tan x is discontinuous function


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