Limits and Continuity of Vector Valued Functions of R

#VectorAnalysis #DifferentialGeometry

Subjects: Vector Analysis
Links: Vector-valued functions of R, Path Connected Sets, Topological Characterization of Continuity in Rn

Def: Let $f : D \subseteq R \to R^{n}$ , and let $a$ be cluster point of $D$ , and $q \in R^{n}$ . Then we can write that $lim_{t \to a} f (t) = q$ if any sequence $(a_{k}) \to a$ , and $a_{k} N e a$ for all $k \in N$ , then $f (a_{k}) \to L$ .

Th: Let $f : D \subseteq R \to R^{n}$ , and $a$ a cluster point of $D$ , then $lim_{x \to a} f (x) = L$ , iff for all $1 \leq i \leq n$ , $lim_{x \to a} f_{i} (x) = q_{i}$ in $R$ .

Th: Let $f : D \subseteq R \to R^{n}$ , and $a$ a cluster point of $D$ , then $lim_{x \to a} f (x) = L$ iff:

\forall ε > 0 \exists > 0 \forall x \in D [0 < | x - a | < δ ⟹ ∥ f (x) - L ∥ < ε]

One Sided Limits

Def: Let $f : D \subseteq R \to R^{n}$ , and let $a$ be a point of $D$ , and $q \in R^{n}$ . Then we can write that $lim_{t \to a^{+}} f (t) = q$ if any sequence $(a_{k}) \to a$ , and $a_{k} > a$ for all $k \in N$ , then $f (a_{k}) \to L^{+}$ .

Th: Let $f : D \subseteq R \to R^{n}$ , and $a$ be a point of $D$ , then $lim_{x \to a^{+}} f (x) = q$ , iff for all $1 \leq i \leq n$ , $lim_{x \to a^{+}} f_{i} (x) = L_{i}^{+}$ in $R$ .

Th: Let $f : D \subseteq R \to R^{n}$ , and $a$ be a point of $D$ , then $lim_{x \to a^{+}} f (x) = L^{-}$ iff:

\forall ε > 0 \exists > 0 \forall x \in D [0 < x - a < δ ⟹ ∥ f (x) - L^{-} | < ε]

Def: Let $f : D \subseteq R \to R^{n}$ , and let $a$ be left cluster point of $D$ , and $q \in R^{n}$ . Then we can write that $lim_{t \to a^{-}} f (t) = L^{-}$ if any sequence $(a_{k}) \to a$ , and $a_{k} < a$ for all $k \in N$ , then $f (a_{k}) \to L^{-}$ .

Th: Let $f : D \subseteq R \to R^{n}$ , and $a$ be a point of $D$ , then $lim_{x \to a^{-}} f (x) = q$ , iff for all $1 \leq i \leq n$ , $lim_{x \to a^{-}} f_{i} (x) = q_{i}$ in $R$ .

Th: Let $f : D \subseteq R \to R^{n}$ , and $a$ a cluster point of $D$ , then $lim_{x \to a^{+}} f (x) = L^{+}$ iff:

\forall ε > 0 \exists > 0 \forall x \in D [0 < a - x < δ ⟹ ∥ f (x) - L^{+} ∥ < ε]

Th: Let $f : D \subseteq R \to R^{n}$ , and $a$ a cluster point of $D$ , then $lim_{x \to a} f (x) = L$ iff, both $lim_{x \to a^{+}} f (x) = L = lim_{x \to a^{-}} f (x)$ .

Def: Let $f : D \subseteq R \to R^{n}$ , and $a \in D$ , $f$ is said to be continuous on $a$ iff $(a_{k}) \subseteq D$ and $a_{k} \to a$ , then $f (a_{k}) \to f (a)$ .

Th: Let $f, g : D \subseteq R \to R^{n}$ , and $a$ be a cluster point of $D$ where $lim_{x \to a} f (x) = L$ and $lim_{x \to a} g (x) = M$ :

$lim_{x \to a} (f + g) (x) = L + M$
$lim_{x \to a} (f \cdot g) (x) = L \cdot M$
Let $c \in R$ , then $lim_{x \to a} (c f) (x) = c L$
Let $ϕ : D \subseteq R \to R$ , $lim_{x \to a} ϕ (x) = k$ then $lim_{x \to a} (ϕ f) (x) = k L$

Def: Let $f : D \subseteq R \to R^{n}$ , and $a \in D$ , $f$ is continuous on $a$ iff for every $1 \leq i \leq n$ , each coordinate function $f_{i}$ is continuous on $a$ .

Cor: Let $f : D \subseteq R \to R^{n}$ , and $a \in D$ , then $f$ is continuous at $a$ iff:

\forall ε > 0 \exists > 0 \forall x \in D [| x - a | < δ ⟹ ∥ f (x) - f (a) ∥ < ε]

Cor: Let Let $f : D \subseteq R \to R^{n}$ , and $a \in D$ , then $f$ is continuous at $a$ iff $lim_{x \to a} f (x) = f (a)$ .

Def: Let $f : D \subseteq R \to R^{n}$ is said to be continuous on $D$ if it is continuous at every $a \in D$ .

Cor: Let $f, g : D \subseteq R \to R^{n}$ and $ϕ : D \subseteq R \to R$ be continuous on $a \in D$ . Then, $ϕ f$ , $f + g, f \cdot g$ are also continuous at $a \in D$ .

Def: Let $f : D \subseteq R \to R^{n}$ , and $a \in D$ , $f$ has a removable discontinuity if $f$ is discontinuous at $a$ , but $lim_{x \to a} f (x) = q \in R^{n}$ . This means $f$ can be modified to be continuous at $a$ by changing its value.