2.2 Continuous Functions and Intervals
Notes
Many common functions are continuous everywhere they are defined.
- Polynomials are continuous on all real numbers.
- Rational functions are continuous except where the denominator is 0.
- $\displaystyle \sqrt{x}$, $e^x$, $\sin x$, $\cos x$ and $\ln x$ are continuous on their domains.
- Sums, products, quotients (with nonzero bottom) and compositions of continuous functions are continuous.
- Example: $\displaystyle f(x) = \frac{x + 1}{x^2 - 4}$ is discontinuous only at $x = 2$ and $x = -2$.
Practice
On what set is $f(x) = x^3 - 5x + 2$ continuous?
Every polynomial is continuous everywhere.
Where is $\displaystyle f(x) = \frac{x + 1}{x^2 - 4}$ discontinuous?
The denominator $x^2 - 4 = (x - 2)(x + 2)$ is 0 at $x = \pm 2$.
What is the largest interval on which $\displaystyle f(x) = \sqrt{x - 3}$ is continuous?
The function is defined and continuous for $x \geq 3$.
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