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I keep seeing this symbol $\equiv$ in Mathematical Analysis -1, Zorich. What does it mean?

For example: in page 180 we have,

Some other pages it occurs in: 117, 139.

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2 Answers

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In this case it means "identically equal" and is a shortcut for saying that a function is defined or that some identity holds for all function values, in contrast to $f(x)=x$ which could also mean a fixed-point equation (that is, $f$ is given and one looks for specific $x$).

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As others have noted, $\equiv$ implies we're saying an identity rather than an equation, i.e. a universal result rather than something to solve. I'm probably not the only one here who feels a bit weird saying identities "aren't equations", so that should probably be an identity rather than a mere equation.

Having said that, $\equiv$ is neither necessary nor sufficient for an identity.

It's unnecessary because, for example, I've never seen anyone bother writing $\sin 2x\equiv 2\sin x\cos x$. In theory we should for clarity; but clarity is more important in some places than others. Zorich probably uses $\equiv$ because you need identities (if only in neighbourhoods) when you differentiate.

It's not sufficient either because you may encounter $\equiv$ to mean an equivalence relation, especially in modulo arithmetic.

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