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Find points at which a given composite function is continuous

by RoRi

Define functions f and g as follows:

    \[ f(x) = \begin{cases} 1 & \text{if } |x| \leq 1 \\ 0 & \text{if } |x| > 1. \end{cases}, \qquad g(x) = \begin{cases} 2-x^2 & \text{if } |x| \leq 2 \\ 2 & \text{if } |x| > 2. \end{cases} \]

Find a formula for h(x) = f(g(x)) and find the values at which h(x) is continuous.

If |x| > 2 then g(x) = 2 and so |g(x)| > 1. Therefore, by the definition of f,

    \[ h(x) = f(g(x)) = 0. \]

If \sqrt{3} < x \leq 2, then |g(x)| = |2 - x^2| > 1 and so again

    \[ h(x) = f(g(x)) = 0. \]

Next, if 1 \leq x \leq \sqrt{3} then |g(x)| = | 2 - x^2| \leq 1 and so

    \[ h(x) = f(g(x)) = 1. \]

Finally, if |x| < 1, then |g(x)| = |2 - x^2| > 1 and we have

    \[ h(x) = f(g(x)) = 0. \]

Putting this all together we have

    \[ h(x) = \begin{cases} 1 & \text{if } 1 \leq |x| \leq \sqrt{3} \\ 0 & \text{otherwise}. \end{cases} \]

From this expression we have h(x) is continuous everywhere except at |x| = 1 and |x| = \sqrt{3}.

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