what's exactly a rigorous definition of
tangent & normal to a curve suitable in all cases?
I know its a dumb qusetion but I have problems
in using it for concepts of calculus.
In fact, there's nothing to stop a line being both a tangent and a normal to the same curve: it could be the tangent at one point, and normal to the curve at another point.
But perhaps that's not what you meant. So see the attachment, which shows the graph of the curve whose equation in polar coordinates is . You'll see that each axis is both a tangent and a normal to the curve at the origin.
I'm not quite sure what you mean byThis curve has a straightforward function that defines it.not to a function