Hi Scholars!
A tricky matter for students first learning analysis is how to choose ε when applying a definition. Here’s an example problem prompt using this with a continuous function.
Prove if a function f is continuous and positive at a point, then there is an interval about that point on which the function is positive.
Let z be the point at which f is positive. By the continuity of the function f, there is positive δ such that all points within δ of z will have function values within ε of f(z). The key is to recognize that, because one is given that f is continuous, they can pick ε as they like. In this case, it suffices to make ε be no more than half the function value at that point (shown in red in diagram). Rearranging this inequality reveals f(x) is positive, and so the function is positive on the interval [z-δ, z+δ].
That’s all for now. Let me know if there are any related examples you think could use some illustrating.
Stay Awesome.
Howard