Statement#

Consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$.

Prove that:

1. $f$ is continuous on $\mathbb{R}$
2. $f$ is not uniformly continuous on $\mathbb{R}$

Required Topics#

• Epsilon-delta definition of continuity
• Uniform continuity
• Sequential characterization of continuity
• Counterexample construction

Definitions#

Continuity at a point: $f$ is continuous at $c$ if for every $\varepsilon > 0$, there exists $\delta > 0$ such that $|x - c| < \delta$ implies $|f(x) - f(c)| < \varepsilon$.

Uniform Continuity: $f$ is uniformly continuous on $A$ if for every $\varepsilon > 0$, there exists $\delta > 0$ such that for all $x, y \in A$, $|x - y| < \delta$ implies $|f(x) - f(y)| < \varepsilon$.

What to Prove#

1. Show continuity at an arbitrary point $c \in \mathbb{R}$
2. Find sequences that violate uniform continuity

Solution#

Solution coming soon.