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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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