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S1.2

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Assume that S⊂ ℝ n S⊂Rn , and that f:S→ ℝ k f:S→Rk is a function. The statement lim x→a f(x)=L limx→af(x)=L is defined to mean that ∀ε>0, ∃δ>0 such that x∈S and 0 0, ∃δ>0 such that x∈S and 0 0,∃x∈S such that 0 0,∃x∈S such that 0<|x−a|<δ. For example, this always holds if a∈ S int a∈Sint , or even if a∈ ( S int ) ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ = a∈(Sint)¯= the closure of S int Sint .

S1.2 1.2: Limits and Continuity $\newcommand{\R}{\mathbb R }$ $\newcommand{\bfa}{\mathbf a}$ $\newcommand{\bfb}{\mathbf b}$ $\newcommand{\bff}{\mathbf f}$ $\newcommand{\bfu}{\mathbf u}$ $\newcommand{\bfx}{\mathbf x}$ $\newcommand{\ep}{\varepsilon}$ Limits and continuity Limits of multivariable functions Continuity Continuous functions and open sets Problems See also Section 1.3 of the textbook. Students should: know the definitions of limit and continuous . know basic properties of limits (Theorems 1-3), of continuity (Theorem 4), and the connection between continuity and open sets (Theorem 5)

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