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Chapter 2: Problem 2

Give the three conditions that must be satisfied by a function to be continuous at a point.

Short Answer

Expert verified
Answer: For a function to be continuous at a point, the following three conditions must be satisfied: 1. The function is defined at the point. 2. The limit of the function exists at the point. 3. The value of the function at the point equals the limit.

Step by step solution

01

Condition 1: Function is Defined at the Point

The function must be defined at the point in question. This means that the point should lie within the domain of the function.
02

Condition 2: Limit Exists at the Point

The limit of the function must exist at the point in question. In mathematical terms, the left-hand limit and the right-hand limit should both exist and be equal at the point. In symbols, for a function f(x) and a point c, \(\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)\).
03

Condition 3: Function Value Equals the Limit

The value of the function at the point should be equal to the limit. In mathematical terms, \(f(c) = \lim_{x \to c} f(x)\). This means that as x approaches the point c, the value of f(x) converges to the value of f(c).

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Most popular questions from this chapter

Graph \(f(x)=\frac{\sin n x}{x},\) for \(n=1,2,3,\) and 4 (four graphs). Use the window \([-1,1] \times[0,5]\) a. Estimate \(\lim _{x \rightarrow 0} \frac{\sin x}{x}, \lim _{x \rightarrow 0} \frac{\sin 2 x}{x}, \lim _{x \rightarrow 0} \frac{\sin 3 x}{x},\) and \(\lim _{x \rightarrow 0} \frac{\sin 4 x}{x}\) b. Make a conjecture about the value of \(\lim _{x \rightarrow 0} \frac{\sin p x}{x},\) for any real constant \(p .\)

Use analytical methods and/or a graphing utility en identify the vertical asymptotes (if any) of the following functions. $$f(x)=\frac{1}{\sqrt{x} \sec x}$$

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Prove the following statements to establish the fact that \(\lim f(x)=L\) if and only if \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L\). a. If \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L,\) then \(\lim _{x \rightarrow a} f(x)=L\) b. If \(\lim _{x \rightarrow a} f(x)=L,\) then \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L\)
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