Let f ∈ C[a, b], and let p be in the open interval (a, b).
a. Suppose f (p) ≠ 0. Show that a δ > 0 exists with f (x) ≠ 0, for all x in [p − δ, p + δ], with [p − δ, p + δ] a subset of [a, b].
b. Suppose f (p) = 0 and k > 0 is given. Show that a δ > 0 exists with |f (x)| ≤ k, for all x in [p − δ, p + δ], with [p − δ, p + δ] a subset of [a, b].