give an algebraic description of a function which satisfies all of the properties simultaneously.?

a) A function f which is continuous on all of R and differentiable on all of R.

(b) A function f which is continuous on all of R except the point x = 5, and which is not
differentiable at x = 5.

(c) A function f which is discontinuous at exactly three different points, has a horizontal
asymptote as x → ∞, but does not have a horizontal asymptote as x → −∞.

(d) A function f with two horizontal asymptotes, and two vertical asymptotes.

(e) A function f which is continuous on all of R, but fails to be differentiable at every integer.