Limits and Continuity

Lesson 5

Objective

Define continuity of functions and determine whether a function is continuous on a particular domain. 

Criteria for Success

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  1. Define continuity of a function $$f$$ at a point $$a$$ as $$f(a)$$ exists, $$\lim_{x\rightarrow a}f(x)$$ exists, and $$\lim_{x\rightarrow a}f(x)=f(a)$$.
  2. Determine whether a function meets the criteria for continuity on a particular domain.
  3. Determine intervals on which a function is continuous.
  4. Sketch functions given constraints on continuity.

Tips for Teachers

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The definition of continuity used in Anchor Problem #1 is a common definition, but the target task uses an alternate definition. Students should build an intuitive understanding of continuity and understand that there are multiple formal ways to describe this idea.

Anchor Problems

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Problem 1

The conditions under which a function $$f$$  is "continuous" at a point $$a$$ are:

A: $$f(a)$$ exists

B: $$\lim_{x\rightarrow a}f(x)$$ exists

C: $$\lim_{x\rightarrow a}f(x) = f(a)$$

 

  1. Sketch a function that meets all three conditions.
  2. Sketch a function that meets conditions A and B but not C.
  3. Sketch a function that meets condition A but not B or C.

Guiding Questions

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Problem 2

The graph below shows $${{f(x)}=|x|}$$. Sketch a slope graph of $${f(x)}$$.

On what interval(s) is the slope graph continuous? What features of a function cause its slope graph to be discontinuous?

Guiding Questions

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Problem Set

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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

  • Include problems generating graphs with different conditions for continuity and constraints from other lessons, including constraints given in interval notation
  • Include practice evaluating limits, and use those limits to find whether a function is continuous
  • Include practice determining on which domains a function is continuous, and writing those domains in interval notation
  • Include problems analyzing slope graphs of different piecewise functions

Target Task

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Below are four conditions. If a point $$a$$ on a function $$f$$ meet these conditions, is it continuous?

A:  $$\lim_{x\rightarrow a^+}f(x)$$ exists

B:  $$\lim_{x\rightarrow a^-}f(x)$$ exists

C:  $$f(a)$$ exists

D:  $$\lim_{x\rightarrow a^+}f(x)=\lim_{x\rightarrow a^-}f(x)=f(a)$$