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--- math104-s21:s:genevievebrooks [2021/05/12 22:17]
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+++ math104-s21:s:genevievebrooks [2026/02/21 14:41] (current)
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-**Continuity**
+**Continuity**\\
+Definitions:
+(Credit to Kaylene Stocking because I really liked the way she described the definitions)\\
+The limit perspective:  Let $(x_n)$ be a sequence of points in the domain of $f$ that converges to some $x_0 \epsilon S$. The sequence $f(xn)$ must converge to $f(x_0)$
+The bounded rate of change perspective:  Pick any point $x_0 \epsilon S$ and any $\varepsilon > 0$. There must exist some $\delta > 0$ so that moving less than $\delta$ away from $x_0$ results in a change of less than $\varepsilon$ in the value of $f(x)$.\\
+**formal definition**: $f$ is continuous at point $p$ if $\forall \varepsilon > 0 \exists \delta > 0$ such that $$d_y(f(x), f(p)) < \varepsilon$$ $$\forall x \epsilon E$$ for which $$d_x(x, p) < \delta$$ Then if f is continuous at every p in its domain, f is continuous overall.
+The topological perspective:  Let E be an open subset of the range of f. The set of points in the domain of f that f maps into E must also be open.

Lecture Notes