Real Analysis: Uniform Continuity and Extreme Value Theorems

Identify all of the extrema on a graph given specific points or intervals, applying the extreme value theorem and recognizing whether the intervals are closed or open.

Using the Extreme Value Theorem, identify the absolute maximum and minimum values of a continuous function defined on a closed interval $[a, b]$.

Prove that the function $f(x) = \frac{1}{x}$ is not uniformly continuous on the interval (0,1).

Prove that the function defined by $f(x) = \frac{1}{1 + x^2}$ is uniformly continuous on $\mathbb{R}$.

Given the function $f(x) = \frac{1}{1 + x^2}$ for all , demonstrate that the supremum is attained at $x = 0$, but the infimum is not attained at any point in $\mathbb{R}$.

If a function $f$ is continuous in the closed interval $[a, b]$, then it is bounded in $[a, b]$.

Prove that the function $x^2$ is uniformly continuous on the closed interval $[-5, 5]$.

Illustrate the difference between continuous functions and uniformly continuous functions using examples.