Continuity on an interval

Continuity on an Interval

Definition:

  • Continuous Function on an Interval: A function f(x) is said to be continuous on an interval [a,b] if it is continuous at every point within the interval [a,b].

Example:

Consider the function f(x)=x

defined on the interval [0,4]. Let's examine its continuity on this interval.

Step 1: Checking f(x) on the Interval [0,4]:

  • At x=0: f(0)=0=0 (Defined)

  • At x=4: f(4)=4=2(Defined)

Step 2: Investigating the Limit at Endpoints:

  • Left-hand Limit at x=0: limx0+f(x)=x

  • , approaching from the right side as x approaches 0.

  • Right-hand Limit at x=4: limx4f(x)=x

  • , approaching from the left side as x approaches 4.

Both these limits exist and are finite.

Step 3: Analysis within the Interval [0,4]:

  • Intermediate Values: For any x between 0 and 4, f(x)=x
  • is defined and continuous.

Conclusion:

The function f(x)=x is continuous on the interval [0,4] as it is defined, and its limits exist and are finite at the endpoints. Moreover, it's continuous for all values within the interval.