Question

Let f:ℝ￫ℝ be continuous on ℝ, and let P:={ x∈ℝ : f(x) > 0 }. If c∈P, show that there exists a neighborhood V_δ (c) ⊆ P

Answer 1

The set P is an inverse image of the open interval (0, +∞). Since f is continuous, it follows from definition that P is open& in ℝ, whence every c from P has an open neighborhood V_c containing in P.