Q.State whether True or False: If $f$ is continuous on its domain $D$, then $|f|$ is also continuous on $D$.

Continuity of Composite Functions

Many functions we meet are really one function fed into another: $\sin(x^2)$ is "square, then take sine"; $\sqrt{x^2 + 1}$ is "add one to the square, then take the root." A natural question is: if each piece is continuous, is the combined function continuous? The answer is a reassuring yes, and it saves an enormous amount of work.

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The Composition Rule

Why it works, in plain terms: as $x \to a$, continuity of $g$ pushes $g(x) \to g(a)$. Then continuity of $f$ at that landing point pushes $f\big(g(x)\big) \to f\big(g(a)\big)$. The limit slides cleanly through both functions:

That last equality is the definition of continuity for $f \circ g$ at $a$.

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Using It in Practice

Most "is this function continuous?" problems become one-liners:

• $\sin(x^2)$ — $x^2$ is continuous everywhere and $\sin$ is continuous everywhere, so the composite is continuous for all $x$.
• $\sqrt{x^2 + 1}$ — the inside is continuous and always $\ge 1 > 0$, and $\sqrt{\ }$ is continuous on $(0, \infty)$, so the composite is continuous everywhere.
• $e^{\cos x}$ — continuous on all of $\mathbb{R}$, being a composition of two everywhere-continuous functions.

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The Trap to Watch