Note: $(f)^2$ means $(f \circ f)$. That is to say, $f$ composed with itself
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To begin this problem, I consider $f(x)=\frac{1}{x}$

For $x \neq 0, f(x)$ is defined

But at $x=0,$ $f(x)=\frac{1}{x}$ has an essential discontinuity since $\lim\_{x \to 0} \frac{1}{x}$ does not exist.

Notice however that $(f)^2=\frac{1}{\frac{1}{x}}=x$ and $\lim\_{x \to 0} x=0$

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This example works when $f$ is disontinuous at a point. How do I construct a function that is discontinuous for **all values** whose composition with itself is continuous for **all values**?