Mississippi Math C.A.13
The Standard
Describe the relationship between differentiability and continuity.
2025 Mississippi College- and Career-Readiness Standards for Mathematics · Use the Definition and Formal Rules of Differentiation to Compute Derivatives
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students decide whether a function is continuous and differentiable at a given point. They explain why differentiability guarantees continuity, but continuity does not guarantee differentiability.
What Mastery Looks Like
- A student can explain that a finite derivative means function values approach the value at that point. For y=|x| at 0, they find left and right slopes of −1 and 1, so no derivative exists.
Common Misconceptions
- Students often assume an unbroken graph has a derivative everywhere, missing corners and vertical tangents. Others mistake a sharp corner for a break in the graph.
How to Assess It
- Exit ticket: At x=0, is y=|x| continuous? Is it differentiable? Give the slope from each side to support your answer.
Lesson moves
Ways to Teach It
Use graph paper and a ruler to draw y=|x| and y=x², then sketch secant lines from both sides toward 0.
Ask, 'Can a graph have no breaks but still lack a derivative? Explain what happens to y=|x| at 0.'
Give students graphs of y=x², y=|x|, and a jump function; have them sort each by continuity and differentiability at 0.
Sketch a bike trail that climbs at 2% then 6% with no height jump; ask whether its grade exists at the join.
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Printable C.A.13 Worksheet
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Related Standards
- A3.G.37
Graph piecewise-defined functions and determine continuity or discontinuities.
- C.A.6
Determine and justify the continuity and discontinuity of functions
- C.A.17
Predict and explain the relationships between functions and their derivatives.
- A3.A.15
Determine asymptotes and holes of rational functions, explain how each was found, and relate these behaviors to continuity.
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