Test 3 Theorems

Sequential Criterion for Functional Limits

Given a function f: A \rightarrow \RR and a limit point c of A, the following are equivalent:
i) \lim_{x \rightarrow c} f(x) = L
ii) For all sequences (x_n) \subseteq A satisfying x_n \neq c and (x_n) \rightarrow c, it follows that f(x_n) \rightarrow L

Algebraic limit theorem functional limits

Let f and g be functions defined on a domain A \subseteq \RR, and assume \lim_{x \rightarrow c} f(x) = L and \lim_{x \rightarrow c} g(x) = M for some limit point c of A. Then
i) \lim_{x \rightarrow c} kf(x) = kL for all k \in \RR
ii) \lim_{x \rightarrow c} [f(x) + g(x)] = L + M
iii) \lim_{x \rightarrow c} [f(x)g(x)] = LM
iv) \lim_{x \rightarrow c} f(x)/g(x) = L/M provided M \neq 0

Divergence Criteria for Functional Limits

Let f be a function defined on A, and let c be a limit point of A. If there exist two sequences (x_n) and (y_n) in A with x_n \neq c and y_n \neq c and \lim x_n = \lim y_n = c but \lim f(x_n) \neq \lim f(y_n), then the functional limit \lim_{x \rightarrow c} f(x) does not exist

Characterizations of Continuity

Let f: A \rightarrow \RR and let c \in A. The function f is continuous at c if and only if any one of the following three conditions is met:
i) for all \epsilon > 0, there exists a \delta > 0 such that |x - c| < \delta (and x \in A) implies |f(x) - f(c)| < \epsilon
ii) For all V_\epsilon(f(c))m there exists V_\delta (c) with the property that x \in V_\delta(c) (and x \in A) implies that f(x) \in V_\epsilon((c))
iii) If (x_n) \rightarrow c (with x_n \in A), then f(x_n) \rightarrow f(c)
iv) If c is a limit point of A, then the above conditions are equivalent to \lim_{x \rightarrow c} f(x) = f(c)

Criterion for Discontinuity

Let f: A \rightarrow \RR, and let c \in A be a limit point of A. If there exists a sequence (x_n) \subseteq A where (x_n) \rightarrow c but such that f(x_n) does not converge to f(c), we may conclude that f is not continuous at c

Algebraic Continuity Theorem

Assue f: A \righarrow \RR are continuous at a point c \in A. Then,
i) kf(x) is continuous at c for all k \in \RR
ii) f(x) + g(x) is continuous at c
iii) f(x)g(x) is continuous at c
iv) f(x)/g(x) is continuous at c, provided the quotient is defined

Composition of Continuous Functions

Let f: A \rightarrow \RR and g: B \rightarrow \RR, assume that the range of f(A) = {f(x): x \in A} is contined in the domain B so that the composition g \circ f(x) = g(f(x)) is defined on A. If f is continuous at c \in A, and if g is continuous at f(c) \in B, then g \circ f is continuous at c