# triple line equals symbol

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.

• This means the function is identically $x$, i.e. we have $f(x)=x$ for all $x$. Similar for $f'$. This is different to $f(x)=x$ which may just mean at some particular $x$ that this is true Commented May 27, 2019 at 20:38

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 $$x$$).
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.