Countable and Uncountable

# Uniform Continuity of Functions Uniform continuity is a stronger form of continuity that applies uniformly over the entire domain of a function. ## Definition A function $f: A \to \mathbb{R}$ is **uniformly continuous** on $A$ if for every $\varepsilon > 0$, there exists a $\delta > 0$ such that for all $x, y \in A$, $$|x - y| < \delta \implies |f(x) - f(y)| < \varepsilon.$$ Unlike regular continuity, $\delta$ depends only on $\varepsilon$, not on the choice of points in $A$. ## Examples ### Example 1: Uniformly Continuous Function The function $f(x) = 2x$ on $\mathbb{R}$ is uniformly continuous. *Proof sketch:* Given $\varepsilon > 0$, choose $\delta = \frac{\varepsilon}{2}$. Then for any $x,y$, $$|x-y| < \delta \implies |f(x)-f(y)| = 2|x-y| < 2 \cdot \frac{\varepsilon}{2} = \varepsilon.$$ ### Example 2: Continuous but Not Uniformly Continuous The function $f(x) = \frac{1}{x}$ on $(0,1)$ is continuous but **not** uniformly continuous. *Reason:* Near zero, the function values change rapidly. No fixed $\delta$ works for all points. --- ## Python Demonstration ```python import numpy as np import matplotlib.pyplot as plt def uniform_continuity_demo(f, domain, name): x = np.linspace(domain[0], domain[1], 500) y = f(x) # Pick pairs with small differences to illustrate uniform continuity diffs = [0.1, 0.05, 0.01] print(f"Checking uniform continuity behavior for {name}") for delta in diffs: max_diff_f = 0 for xi in x: # Check points xi and xi+delta if in domain if xi + delta <= domain[1]: diff_f = abs(f(xi) - f(xi + delta)) max_diff_f = max(max_diff_f, diff_f) print(f"For delta = {delta}, max |f(x)-f(x+delta)| = {max_diff_f:.4f}") plt.plot(x, y, label=name) plt.title(f"Function: {name}") plt.xlabel("x") plt.ylabel("f(x)") plt.grid() plt.legend() plt.show() # Example 1: f(x) = 2x on [-5,5] uniform_continuity_demo(lambda x: 2*x, (-5, 5), "f(x) = 2x") # Example 2: f(x) = 1/x on (0.01, 1) uniform_continuity_demo(lambda x: 1/x, (0.01, 1), "f(x) = 1/x")