- Determine whether a function is continuous at a number.
- Determine the input values for which a function is discontinuous.
Parking Garage Pricing
Most functions in real life follow a piecewise pattern with jump discontinuities depending on the conditions.
A downtown parking garage charges $4 per hour or any fraction of an hour, with a $24 daily maximum. The cost function is:
[latex]C(t) = \begin{cases} 4 & 0 < t \leq 1 \\ 8 & 1 < t \leq 2 \\ 12 & 2 < t \leq 3 \\ 16 & 3 < t \leq 4 \\ 20 & 4 < t \leq 5 \\ 24 & t > 5 \end{cases}[/latex]
where [latex]t[/latex] is hours parked and [latex]C(t)[/latex] is cost in dollars.
This function is not continuous on the hour as the price “jumps” from one level to the next.
How To: Determining Continuity at a Point
- Check Condition 1: Does [latex]f(a)[/latex] exist?
- Check Condition 2: Does [latex]\lim_{x \to a} f(x)[/latex] exist? (Do left and right limits equal each other?)
- Check Condition 3: Does [latex]\lim_{x \to a} f(x) = f(a)[/latex]?
- If all three conditions hold, the function is continuous at [latex]x = a[/latex]
- If conditions fail, identify the type of discontinuity
A laboratory temperature control system is modeled by:
[latex]T(t) = \begin{cases} 4t & t \leq 3 \\ 8 + t & t > 3 \end{cases}[/latex]
where [latex]t[/latex] is time in hours and [latex]T(t)[/latex] is temperature in degrees Celsius. Determine whether [latex]T(t)[/latex] is continuous at [latex]t = 3[/latex].
Is [latex]E(x)[/latex] continuous at [latex]x = 3[/latex]? [response area] (yes or no)
A data transmission rate is:
[latex]T(x) = \begin{cases} \sin(x) & x < 0 \ x^3 & x > 0 \end{cases}[/latex]
where [latex]x[/latex] is signal strength. What type of discontinuity exists at [latex]x = 0[/latex]?
Type of discontinuity at [latex]x = 0[/latex]: [response area]