Define what it means to say that a function f : R → R is continuous at c ∈ R. Let f : R → R and g : R → R be functions that are continuous at 0, with f(0) > g(0). Prove that there is a δ > 0 such that f(x) > g(x) for all x ∈ (−δ, δ).
so far this is what I have done: A function f : R → R is continuous at a point c ∈ R if, for every ε > 0, there exists a δ > 0 such that:|x − c| < δ ⇒ |f(x) − f(c)| < ε.
let ε>0. there exists δ1, δ2 > 0 such that:|x| < δ1 ⇒ |f(x) − f(0)| < ε and |x| < δ2 ⇒ |g(x) − g(0)| < ε.
Taking δ = min{δ1, δ2}, then |x| < δ implies that: f(x) > f(0) − ε = g(0) + ε > g(x)
my question is, I do not know how to find ε to finish off the proof.
The equality $f(0)-\epsilon=g(0)+\epsilon$ holds only for the choice $\epsilon=\frac{f(0)-g(0)}{2}$. This is positive since $f(0)>g(0)$.