difference between equivalent and equal sign

Question

I am reading a statement in which we have to show that $f(x)$ is equivalent to $0$ on an interval.

What is differences $f(x)$ is $0$ and $f(x)$ is equivalently $0$ on an interval.

Please help.

Thanks

Answer 1

People write $f(x)\equiv0$ on an interval to mean that the function is identically $0$ on the interval - that is the only value $f(x)$ takes is on the interval is $0$. This notation is sometimes used to avoid confusion that arises with $f(x)=0$ which looks like one is attempting to solve the question which $x$ satisfy $f(x)=0$.