A function can have a value at a point without being continuous there. It can also be continuous without having a derivative. Work through those distinctions before applying a differentiation rule to a piecewise definition.
Curriculum checked 2 October 2026 · original practice for selected skills.
You will need: Substitution, simple limits and the derivative of a polynomial.
Attempt each task before opening a hint. If you need help, reveal the first hint, then the setup, then compare your written steps with the solution. Finish with the retests without referring back.
Continuity at a point requires the left limit, right limit and assigned function value to agree.
Differentiability compares the limiting slopes from the two sides. A sharp corner can be continuous but fail this test.
Differentiability implies continuity. The reverse implication is false, so a continuity check cannot replace a derivative check.
The rule containing equality gives f(2).
Match 2k + 1 to 7.
The left limit is 2k+1. The right limit and value are 7.
Solve the equality.
Answer: k = 3
Before moving on: can you explain why your method works, as well as give the answer?
Write a method as well as an answer. The tasks change the reasoning, not just the numbers.
Factor before substituting.
Away from 3, the expression equals x+3.
Cancel the factor only for x≠3.
The assigned value must equal that limit.
Answer: c = 6
Before moving on: can you explain why your method works, as well as give the answer?
Write separate rules for negative and positive x.
The left rule is −x and the right rule is x.
Both limits and f(0) are zero, so it is continuous.
The left derivative is −1 and the right derivative is +1.
Answer: Continuous, but not differentiable at 0.
Before moving on: can you explain why your method works, as well as give the answer?
There is an outer power and an inner linear function.
Differentiate the outside, then multiply by the derivative of 3x+2.
Bring down the power and reduce it by one.
Multiply by 3.
Answer: 12(3x+2)³
Before moving on: can you explain why your method works, as well as give the answer?
A tempting claim: “A continuous function must be smooth.”
Continuity controls values near the point, not whether the left and right slopes agree.
These changed questions test whether you can reconstruct the method. If you use a hint, record where you got stuck and retry later. Completing this small set does not establish full chapter mastery.
Match the left limit to the assigned value.
a−2=4.
The left limit is a−2; the other side is 4.
Add 2 to both sides.
Answer: a = 6
Could you solve this without the earlier example? If not, revisit the first line where you got stuck.
Apply the chain rule.
Multiply 5(2x−1)⁴ by 2.
The new power is 4.
The inner derivative is 2.
Answer: 10(2x−1)⁴
Could you solve this without the earlier example? If not, revisit the first line where you got stuck.
If a retest exposed the same error, rewrite the first incorrect step and explain its correction aloud. If both were independent, return to a mixed exercise or a missed paper question. A parent can ask what changed in the method rather than only asking for the answer.
At a joining point, check continuity first. If it fails, differentiability fails there too. Away from the join, use the rule for that interval.
Yes. Use the two print buttons above. The question sheet leaves working space and hides solutions; the worked-solutions option includes the solution steps. Your browser can save either view as a PDF.
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