Continuity at the left endpoint of a closed interval uses the right-hand limit available within that interval.

At an interior point 2, f(2) = 9 and the finite limit is 7. Which continuity requirement fails?

• The function value must be defined.
• The finite limit must exist.
• The limit must equal the value.
• The one-sided limits must be finite.

Which condition is NOT required for continuity at the interior point 3?

• f(3) is defined
• The finite limit at 3 exists
• The derivative at 3 exists

What kind of discontinuity does 1/x have at 0?

• Infinite
• Removable
• Jump

Let f(x) = 2x below 3 and f(x) = x+k at or above 3. Which k makes f continuous at 3?

• 6
• 3
• 0
• 9

For f(x) = x²-2 on [1,2], f(1) = -1 and f(2) = 2. At least how many roots does the Intermediate Value Theorem guarantee in (1,2)?

At -1, a function has finite limit 4 but assigned value 10. Which replacement value makes it continuous there?

• 4
• 10
• 6
• 14

A continuous function on [0,2] has f(0) = -2, f(1) = 3 and f(2) = -1. At least how many distinct interior roots are guaranteed?

For f(x) = x(1-x) on [0,1], both endpoint values are zero and f is positive inside. Which conclusion is correct?

• Two endpoint roots force a third root inside.
• The endpoint roots do not establish an interior root.
• The polynomial is discontinuous because its endpoint values agree.
• The Intermediate Value Theorem forces zero at the midpoint.

For f(x) = x³+x-1, f(0) = -1 and f(1) = 1. What justifies an interior root?

• Polynomial continuity and opposite endpoint signs.
• The degree alone fixes an exact root between the endpoints.
• Opposite signs locate the root at the interval's midpoint.