How does constant frequency response relate to equiripple?
What your teacher tried to show is that a transfer function which is the ratio of two polynomials can never be constant (excluding the trivial case where both polynomials are of degree zero). This means that if your desired frequency response is a piecewise constant function (as is the case for an ideal frequency selective filter), you can only approximate that piecewise constant function by a realizable transfer function, and one popular approximation is an equi-ripple approximation, where the maximum deviation from the desired constant value is minimized.