Continuity
Yuanting Lu
09/02/2020
Types of Discontinuity
- A removable discontinuity
- A jump discontinuity
- A infinite discontinuity
- An essential discontinutiy
Example 1. Identify the locations and types of discontinuities of the function whose graph is shown below.
Example 2. Identify the locations and types of discontinuities of the function \[f(x) = \frac{x^2 + x - 2}{x^2 - x - 2}.\]
Example 3. What types of discontinuity do you find in the following functions?
- \(f(x) = \dfrac{x - 3}{x^2 - 9}\)
- \(f(w) = \dfrac{w^2 + w -12}{|w - 3|}\)
- \(f(x) = \dfrac{\sqrt{2-x} - \sqrt{2 + x}}{x}\)
- \(f(x) = \dfrac{\frac 1 3 - \frac 1 x}{x - 3}\)
Continuity at a Point
Suppose \(a\) is in the domain of a function \(f\). Then \(f\) is
Continuous at \(a\) if \(\displaystyle\lim_{x\to a} f(x)=f(a)\).Continuous from the right at \(a\) if \(\displaystyle\lim_{x\to a^+} f(x)=f(a)\).Continuous from the left at \(a\) if \(\displaystyle\lim_{x\to a^-} f(x)=f(a)\).
Continuity on an Interval
A function is
continuous on an interval if it is continuous at every point in the interior of that interval.A function \(f\) is
continuous if it is continuous on every interval in its domain.
polynomial, rational, root, exponential, logarithmic, trigonometric, and inverse trigonometric functions are all continuous on their domains.
The sum, difference, product, quotient, and
Example 4. Determine the domain of continuity of \(f(x) = \sin(e^x)\).
Example 5. Determine the domain of continuity of the Heaviside function
\[H(x)=\left\{ \begin{array}{ll} 0, & x < 0 \\ \frac{1}{2}, & x = 0 \\ 1, & x>0 \end{array}\right. \]
Example 6. Consider \(f(x)=\ln\big(H(x-2)\big)\). Determine its domain and its domain of continuity.
Example 7. Find the value(s) of the constant \(c\) that make the piecewise define function continuous. \[ \displaystyle \left\{ \begin{array}{ll} x^2+cx-1, & x \ge c\\ cx+3, & x < c \end{array} \right. \]
Intermediate Value Theorem
Suppose the function \(f(x)\) is continuous on the closed interval \([a,b]\) and satisfies \(f(a)\ne f(b)\). Let \(y\) be any number between \(f(a)\) and \(f(b)\). Then there exists a number \(c\) such that \(a< c <b\) and \(f(c)= y\).
Example 8. Suppose a function \(f\) is continuous on the interval \([-3,4]\). And suppose \(f(-3) = 1\) and \(f(4) = -1\). Must \(f\) have a zero in that interval?
Example 9. Use the Intermediate Value Theorem to show that the equation \(\sin x = x^2 - x\) has at least one real root on the interval \((1, 2)\).