Florida B.E.S.T. MA.912.C.2.9
The Standard
Demonstrate and use the relationship between differentiability and continuity.
Florida B.E.S.T. Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students determine whether a function is continuous and differentiable at a given point using graphs, formulas, and limits. They show that differentiability guarantees continuity, but continuity alone does not guarantee differentiability.
What Mastery Looks Like
- Students correctly identify where a function is continuous and where it is differentiable. They explain why an existing derivative guarantees continuity and use f(x) = |x| as a counterexample to the converse.
Common Misconceptions
- Students often assume every continuous function is differentiable. They may also miss corners, cusps, and vertical tangents because the graph has no break. Some think a function can be differentiable at a jump or hole.
How to Assess It
- Give students the graph of f(x) = |x| and ask: “Is the function continuous at x = 0? Is it differentiable there? Justify both answers.”
Lesson moves
Ways to Teach It
Sort graph cards into four labeled groups based on whether each function is continuous and differentiable at the marked point.
Write a response to: “If a graph has no breaks, must it have a derivative everywhere?” Include a counterexample.
Play a partner card match where students pair each graph with its continuity and differentiability description, then defend each match.
Analyze a car’s continuous position-time graph with a sharp corner, and explain why its instantaneous velocity is undefined there.
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Related Standards
- MA.912.C.1
Develop an understanding for limits and continuity. Determine limits and continuity.
- MA.912.C.2
Develop an understanding for and determine derivatives.
- MA.912.C.1.9
Define continuity in terms of limits.
- MA.912.C.1.10
Given the graph of a function, identify whether a function is continuous at a point. If not, identify the type of discontinuity for the given function.
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