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 $h(x) = f(g(x))$ . Find the values at which $h(x)$ is continuous.

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

$h(x) = f(g(x)) = f(x) = \frac{x-x}{2} = 0.$

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

$h(x) = f(g(x)) = f(x^2) = \frac{2x^2}{2} = x^2.$

Putting these together,

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

Certainly $h$ is continuous for all $x \neq 0$ since the constant function $h(x) =0$ and the polynomial $h(x) = x^2$ are continuous. Further, $h$ is continuous at $x = 0$ since

$\lim_{x \to 0^-} h(x) = \lim_{x \to 0^+} h(x) = \lim_{x \to 0} h(x) = 0 = h(0).$

Stumbling Robot

Find points at which a given composite function is continuous

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