Continuity at a point

Continuity at a Point

Definition:

  • Continuity at a Point c: A function f(x) is said to be continuous at a point c in its domain if three conditions are met:
    1. f(c) is defined.
    2. The limit of f(x) as x approaches c exists.
    3. The limit of f(x) as x approaches c is equal to f(c).

Example:

Consider the function f(x)=x21x1.

Let's investigate the continuity of f(x) at x=1.

Step 1: Checking if f(1) is Defined:

f(1)=12111=00 (undefined)

Step 2: Evaluating the Limit limx1f(x):

limx1f(x)=limx1x21x1

Using factorization or canceling out the common factor (x1):

=limx1(x+1)(x1)x1

=limx1(x+1)=2

Step 3: Comparing f(1) and limx1f(x):

f(1) is undefined (as 00), but limx1f(x)=2.

Since f(1) is undefined, and the limit of f(x) as x approaches 1 exists and is equal to 2, the function is discontinuous at x=1.

Conclusion:

The function f(x)=x21x1 is not continuous at x=1 due to the undefined value of f(1), even though the limit as x approaches 1 exists and is equal to 2