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Find points at which a given composite function is continuous

Define functions f and g as follows:

    \[ f(x) = \frac{x+|x|}{2}, \text{ for all } x, \qquad g(x) = \begin{cases} x & \text{for } x < 0,\\ x^2 &\text{for } x \geq 0. \end{cases} \]

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


If x < 0, we have |x| = -x and we compute the composition,

    \[ h(x) = g(f(x)) = g \left( \frac{x - x}{2} \right) = g(0) = 0. \]

If x \geq 0, then |x| = x and we compute the composition,

    \[ h(x) = g(f(x)) = g \left( \frac{x+x}{2} \right) = g(x) = x^2. \]

Hence,

    \[ h(x) = \begin{cases} 0 & \text{if } x < 0, \\ x^2 & \text{if } x \geq 0. \end{cases} \]

This is the same function as in this previous exercise, and it is continuous everywhere.

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