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# Continuity

If there are no holes in your function, it's continuous! Many powerful theorems in Calculus only apply to these special types of functions.

# Continuity Warmup

Which of the graphs show a function that is continous at $$x=a?$$

$f(x) = \frac{x^2}{(x-1)(x-2)}$

At how many $$x$$-values is this function discontinuous?

$f(x)=\frac { x }{ x } ,g(x)=\frac { x^{ 2 } }{ x } ,h(x)=\frac { x }{ x^{ 2 } }$

Each of these functions is undefined at $$x = 0.$$ Which functions is it possible to extend by defining a functional value at $$x = 0$$ in such a way that the resulting extended function is continuous at $$x = 0?$$

True or False?

The function $$f(x) = x^3 + x - 1$$ has a root between 0 and 1.

  

Hint: $$f(0) = -1$$ and $$f(1) = 1.$$ Apply the Intermediate Value Theorem.

Which of the graphs shows a function $$f(x)$$ for which $$\lim_{x\to a} f(x)$$ exists?

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