Explanation
Solution
A function is continuous at x=a if:
x→a−limf(x)=x→a+limf(x)=f(a)
1. Find the Left-Hand Limit (LHL)
For x < 2, we use f(x)=25−x:
2. Find the Right-Hand Limit (RHL)
For x > 2, we use f(x)=x−23:
3. Find the Value of the Function at x=2
From the given definition, when x=2:
Conclusion
Comparing the results:
Since the LHL=RHL, the limit exists and is equal to 21. However, the limit value (21) is not equal to the function value (1).
Therefore, the function is discontinuous at x=2. This is known as a removable discontinuity.
Correct Option: (B)