  In this paper, we introduce the class of $(\beta,\gamma)$-Chebyshev functions
and corresponding points, which can be seen as a family of {\it generalized}
Chebyshev polynomials and points. For the $(\beta,\gamma)$-Chebyshev functions,
we prove that they are orthogonal in certain subintervals of $[-1,1]$ with
respect to a weighted arc-cosine measure. In particular we investigate the
cases where they become polynomials, deriving new results concerning classical
Chebyshev polynomials of first kind. Besides, we show that subsets of Chebyshev
and Chebyshev-Lobatto points are instances of $(\beta,\gamma)$-Chebyshev
points. We also study the behavior of the Lebesgue constants of the polynomial
interpolant at these points on varying the parameters $\beta$ and $\gamma$.
