How do you see this function is odd?

Question

I thought I am supposed to do check $f(t)=-1$ and compare with $f(-t)$

$f(-t)=-1$

If $f(t)=f(-t)$ the function is even. But this function is odd.

Answer 1

It is for sure not odd function as f(0)=-1. For odd f(0)= 0 only.

Answer 2

You should specify that you are adopting the convention that a discontinuous function of the first kind assumes in the discontinuity point the average of the left and right limits;

In that way, left limit is -1, right limit is +1, the average is 0 and the function then is odd as

$$f(x)=1\quad(x>0)$$ $$f(-x)=-1=-f(x)$$ $$f(0)=0$$

Anyway, the definition of odd function only wants $$f(-x)=-f(x)$$ So if the function is undefined at x=0 there’s no point in insisting on f being 0 there

See for reference: Do odd functions pass through the origin?