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

fis 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