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