Fundamental Theorem Of Calculus

Step 1: Understand the discontinuity at .
A function is continuous at if . Discontinuity means this fails: either the limits differ, the limit doesn’t equal , or the limit doesn’t exist.
Step 2: Analyze the options.
Option 1: Implies continuity, which contradicts the problem.
Option 2: The left-hand limit equals , but if the right-hand limit differs, the function is discontinuous (e.g., jump discontinuity).
Options 3 and 4: Unrelated to discontinuity at , as they involve and .
Step 3: Conclusion.
Option 2 allows for discontinuity (e.g., if ), making it the correct choice.

Step 1: Understand what makes a function decreasing.
A function is decreasing if . Interval: , where , .
Step 2: Analyze each function.
Option 1: , . In , , so . Increasing.
Option 2: , . Changes sign at , not decreasing throughout.
Option 3: , . Increasing.
Option 4: . Let , , so . Decreasing.
Step 3: Conclusion.
Only is decreasing over the entire interval.