Microscopic tangency in one variable
One important comment about the proof of the Extreme Value Theorem is this. The simple fact that the standard part of every hyperreal satisfying is in the original real interval is the form that topological compactness takes in Robinson's theory: A standard topological space is compact if and only if every point in its extension is near a standard point, that is, has a standard part and that standard part is in the original space.
Suppose and are real functions defined on the interval , if we know that for all hyperreal numbers with and , with whenever , then arguments like the proof of the simple equivalency of limits and infinitesimals above show that is a uniform limit of the difference quotient functions on compact subintervals . More generally, we can show:
Theorem: Uniform Differentiability
Suppose and are real functions defined on the open real interval . The following are equivalent definitions of, "The function is smooth with continuous derivative on ."