Venn Diagrams

To determine which statement about the function is true, we explore its continuity based on the given piecewise definition: . Below, we analyze its continuity at the points where the function definition changes, namely :

1. Continuity at :

• Left-hand Limit (LHL) as : .
• Right-hand Limit (RHL) as : .
• Limit Condition: Continuity requires , implying .

2. Continuity at :

• LHL as : .
• RHL as : since .
• Limit Condition: Continuity at requires .

3. Continuity at :

• LHL as : .
• RHL as : since , and hence becomes .
• Limit Condition: Continuity at requires .

To be discontinuous at exactly one point, one limit condition must fail. Suppose only at we have a discontinuity:

• This implies not all conditions for continuity are satisfied simultaneously.
• Solving previous conditions with one failure, differing from 1 hints , and fulfilling tested continuity relation: not resolving as consistently.

After analyzing potential discontinuities of , if is discontinuous at exactly one point, then the condition is valid.