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--- math104-s21:s:ryanpurpura [2021/05/12 23:25]
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+++ math104-s21:s:ryanpurpura [2026/02/21 14:41] (current)
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 : Give an example where it's neither.
-A: Consider $E = \{1, 1/2, 1/3, 1/4, \dots}$.
+A: Consider $E = \{1, 1/2, 1/3, 1/4, \dots \}$.
 $E' = \{0\}$. But this is neither subset nor superset of $E$.
+: Why is continuity defined at a point and uniform continuity defined on an interval?
+A: Continuity allows you to pick a different $\delta$ for each point,
+allowing you to have continuity at a single point. Uniform continuity
+needs to have a interval share the same value of $\delta$.
+: What is a smooth function, and give an example one.
+A: A smooth function is infinitely differentiable.
+All polynomials are smooth.
+: Prove differentiability implies continuity.
+A: Differentiability
+implies $\lim_{t \to x} \frac{f(t) - f(x)}{t-x} = L$.
+Now multiply both sides by $\lim_{t \to x} (t -x)$, taking advantage of limit rules.
+so we get $\lim_{t\to x} \frac{f(t) -f(x)}{t-x}(t-x) = f'(x) \cdot 0 = 0$
+which means $\lim_{t\to x} f(t) = f(x)$.
+: Use L'Hospital's Rule to evaluate $\lim_{x\to \infty} \frac{x}{e^x}$.
+A: We clearly see that the limits of the top and bottom go to infinity,
+so we can use L'Hospital's Rule.
+Taking derivative of top and bottom, we get $1/e^x$ which goes to $0$.
+: 5.1 Prove that if $c$ is an isolated point in $D$, then $f$ is automatically continuous at $c$.
+A: Briefly, no matter what $\epsilon$ is chosen, we can just choose a small
+enough neighborhood such $c$ is within the neighborhood.
+Then all $x$ (i.e., just c) in the neighborhood have $|f(x)-f(c)| = 0$.

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