Calculus
# Continuity

Which of the graphs show a function that is continous at \(x=a?\)

\[g(x) = \begin{cases} \frac{1}{x} & \textrm{if } x \neq 0 \\ a & \textrm{if } x = 0 \\ \end{cases}\]

Is there a real number \(a\) for which the function \(g(x)\) is continuous at \(x = 0?\)

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

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.

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**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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