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1.12 Confirming Continuity over an Interval

Keywords

Interval Continuity 区间连续性 • Domain 定义域

Endpoint Continuity 端点连续性 • Catalog of Functions 函数类别

Continuity on an Interval

A function is continuous on an interval if its graph can be drawn without lifting the pencil anywhere within that range.

核心要点：Continuity on an Interval

  1. Open Interval \((a, b)\): A function \(f\) is continuous on \((a, b)\) if it is continuous at every point in the interval.

  2. Closed Interval \([a, b]\): A function \(f\) is continuous on \([a, b]\) if:

    • It is continuous on the open interval \((a, b)\).
    • It is continuous from the right at \(a\): \(\lim\limits_{x \to a^+} f(x) = f(a)\).
    • It is continuous from the left at \(b\): \(\lim\limits_{x \to b^-} f(x) = f(b)\).

The Catalog of Continuous Functions

One of the most important tools in Calculus is knowing which functions are "naturally" continuous. According to LIM-2.B.2, the following functions are continuous at every point in their domains:

  • Polynomials: (e.g., \(x^3 - 2x + 1\)) — Continuous on \((-\infty, \infty)\).
  • Rational Functions: (e.g., \(\frac{P(x)}{Q(x)}\)) — Continuous everywhere except where the denominator \(Q(x) = 0\).
  • Power Functions: (e.g., \(x^{2/3}\), \(\sqrt{x}\)) — Continuous on their defined domains.
  • Exponential & Logarithmic: (e.g., \(e^x\), \(\ln x\)) — \(e^x\) is continuous on \((-\infty, \infty)\); \(\ln x\) is continuous on \((0, \infty)\).
  • Trigonometric Functions: (e.g., \(\sin x\), \(\cos x\), \(\tan x\)) — \(\sin\) and \(\cos\) are continuous everywhere; \(\tan x\) is continuous except at its vertical asymptotes.

Example: Continuity of the Square Root

Consider \(f(x) = \sqrt{x}\). Its domain is \([0, \infty)\).

Is it continuous at the endpoint \(x=0\)? Yes, because \(\lim\limits_{x \to 0^+} \sqrt{x} = 0\), which matches \(f(0)\). Therefore, we say \(f(x) = \sqrt{x}\) is continuous on its entire domain \([0, \infty)\).

Extension: Continuity vs. Differentiability

While every function we can differentiate is continuous, not every continuous function can be differentiated. A classic example is \(f(x) = |x|\). It is perfectly continuous on \((-\infty, \infty)\) (you never lift your pencil), but it has a "sharp corner" at \(x=0\) where the slope is undefined. We will explore this further in Unit 2!

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