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

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

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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.

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

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

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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?

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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

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