Continuity
Picture a hike up a hill in Yellowstone: your elevation changes smoothly, and a split second later you are almost at the same height. Now picture a downtown parking garage that charges by the hour: at 1 hour 59 minutes you pay one price, and at exactly 2 hours the price jumps. The hike is continuous; the garage fee is not. In this section you will turn that "no jumps" intuition into a precise test using limits, and then use it to solve equations, find missing constants, and prove that solutions exist.
Continuity at a point and on a domain
A function is continuous at $x=a$ if its graph does not break there: the value the function approaches from both sides matches the value it actually has at $a$.
You do not have to run the three-step test for every function. Most functions you know are built from pieces that are continuous wherever they are defined.
A function is continuous on an interval if it is continuous at every point of that interval. At a closed endpoint, only the limit from inside the interval is used.
Determine the domain of the function $g(x) = \sqrt{x^2 - 25}$ and justify in which intervals the function is continuous.
Determine the domain of the function $g(x) = \sqrt{x^2 - 25}$ and justify in which intervals the function is continuous.
Types of discontinuity
When one of the three conditions fails at $x=a$, the function is discontinuous there. Three types show up again and again.
To read a graph: an open circle with a filled dot elsewhere (or nothing) signals a hole; two branches ending at different heights signal a jump; branches that shoot up or down along a vertical line signal an infinite discontinuity.
For rational functions there is a fast rule. Factor the numerator and denominator. A factor that cancels gives a hole (removable). A factor that stays in the denominator gives a vertical asymptote (infinite).
Observe the following graphical representation of a function $f(x)$ and determine the points where it presents discontinuities, classifying them into removable, finite…
Observe the following graphical representation of a function $f(x)$ and determine the points where it presents discontinuities, classifying them into removable, finite jump or infinite jump.
Analyze the continuity of the function $f(x) = \frac{x^2 - 16}{x - 4}$. State at which points it is discontinuous and classify the type of discontinuity it presents.
Analyze the continuity of the function $f(x) = \frac{x^2 - 16}{x - 4}$. State at which points it is discontinuous and classify the type of discontinuity it presents.
Analyze the continuity of the rational function $f(x) = \frac{2x + 1}{x^2 + x - 6}$. Find the points where it is not continuous and specify the type of discontinuity in…
Analyze the continuity of the rational function $f(x) = \frac{2x + 1}{x^2 + x - 6}$. Find the points where it is not continuous and specify the type of discontinuity in each case.
Piecewise functions
If each piece of a piecewise function is continuous on its own interval, the only new places to check are the break points, where the formula changes. At those points you compare one-sided limits.
Absolute value is piecewise. Recall $|u| = u$ when $u\ge 0$ and $|u| = -u$ when $u<0$. So $|x+1|$ is the piecewise function $x+1$ for $x\ge -1$ and $-(x+1)$ for $x<-1$. At $x=-1$ both pieces equal $0$, and so does $f(-1)$. Therefore $|x+1|$ is continuous at $x=-1$ (it has a corner there, but no break).
Step functions. The greatest integer function $\lfloor t\rfloor$ gives the largest integer that is $\le t$. For example, $\lfloor 1.99\rfloor = 1$ and $\lfloor 2\rfloor = 2$. It jumps at every integer. (Some problems write it as $\text{ENT}(t)$, "integer part".)
Given the piecewise function: $f(x) = \begin{cases} 3x - 2 & \text{if } x \le 2 \\ x^2 & \text{if } x > 2 \end{cases}$ Determine analytically whether the function is…
Given the piecewise function:
$f(x) = \begin{cases} 3x - 2 & \text{if } x \le 2 \\ x^2 & \text{if } x > 2 \end{cases}$
Determine analytically whether the function is continuous at the point $x = 2$.
Express the function $h(x) = |x - 3|$ as a piecewise function and prove analytically that it is continuous at $x = 3$.
Express the function $h(x) = |x - 3|$ as a piecewise function and prove analytically that it is continuous at $x = 3$.
The fee for a public parking lot is calculated using the function $C(t) = 2 + 1.5 \cdot \text{ENT}(t)$, where $t$ is the time in hours and $\text{ENT}(t)$ represents the…
The fee for a public parking lot is calculated using the function $C(t) = 2 + 1.5 \cdot \text{ENT}(t)$, where $t$ is the time in hours and $\text{ENT}(t)$ represents the integer part function of $t$.
Identify the points of discontinuity of this function in the interval $(0, 4)$ and explain the real meaning of these discontinuities for a parking lot user.
Finding parameters that make a function continuous
A classic problem: a piecewise function has an unknown constant, and you must find the value that removes the jump.
With two unknowns you need two equations, one for each break point.
When a piece contains $\ln x$ or $e^{x-c}$, remember the key values $\ln 1=0$, $\ln e=1$, $e^0=1$. They often make the substitution easy.
Find the value of the parameter $k$ so that the following function is continuous throughout its domain:…
Find the value of the parameter $k$ so that the following function is continuous throughout its domain:
$f(x) = \begin{cases} \frac{x^2 - 9}{x + 3} & \text{if } x \ne -3 \\ k & \text{if } x = -3 \end{cases}$
Calculate the value of $a$ so that the function $f(x)$ is continuous at $x = 1$:…
Calculate the value of $a$ so that the function $f(x)$ is continuous at $x = 1$:
$f(x) = \begin{cases} ax^2 - 3 & \text{if } x \le 1 \\ e^{x-1} + 2a & \text{if } x > 1 \end{cases}$
Determine the values of the parameters $a$ and $b$ so that the following function is continuous throughout $\mathbb{R}$:…
Determine the values of the parameters $a$ and $b$ so that the following function is continuous throughout $\mathbb{R}$:
$f(x) = \begin{cases} ax^2 + bx + 1 & \text{if } x \le 1 \\ 2a \ln(x) + b & \text{if } 1 < x \le e \\ e^{x-e} + a & \text{if } x > e \end{cases}$
One-sided limits near vertical asymptotes
To classify a discontinuity you sometimes need to know what a formula does as $x$ approaches a number from only one side. The building block is $\dfrac{1}{x-a}$.
Now feed that into another function. For $e^u$: if $u\to+\infty$ then $e^u\to+\infty$, and if $u\to-\infty$ then $e^u\to 0$.
Piecewise functions can include a branch like this. If a branch heads to $\pm\infty$ at a break point, the one-sided limit does not exist as a finite number, and the discontinuity is infinite.
Another handy fact: $\dfrac{|u|}{u}$ equals $1$ when $u>0$ and $-1$ when $u<0$ (because $|u|=u$ or $-u$). So $\dfrac{|x-2|}{x-2}$ is $-1$ for $x<2$ and $1$ for $x>2$. Its one-sided limits at $2$ are $-1$ and $1$: a jump of size $2$.
Given the exponential function $g(x) = e^{\frac{1}{x-1}}$, study its continuity at the point $x = 1$ by calculating the one-sided limits.
Given the exponential function $g(x) = e^{\frac{1}{x-1}}$, study its continuity at the point $x = 1$ by calculating the one-sided limits.
Check if the following function is continuous at $x = 0$: $f(x) = \begin{cases} \frac{1}{x} & \text{if } x < 0 \\ x^2 + 1 & \text{if } x \ge 0 \end{cases}$
Check if the following function is continuous at $x = 0$:
$f(x) = \begin{cases} \frac{1}{x} & \text{if } x < 0 \\ x^2 + 1 & \text{if } x \ge 0 \end{cases}$
Find the values of $a$ and $b$ that make the following function, which includes an absolute value, continuous throughout its domain:…
Find the values of $a$ and $b$ that make the following function, which includes an absolute value, continuous throughout its domain:
$f(x) = \begin{cases} \frac{|x - 2|}{x - 2} + ax & \text{if } x < 2 \\ b & \text{if } x = 2 \\ x^2 - ax + 1 & \text{if } x > 2 \end{cases}$
Algebra tools for $\frac{0}{0}$ limits
Many continuity problems end with a limit that is $\frac{0}{0}$ when you substitute. Three tools handle almost all of them.
-
Factor and cancel (polynomials, rational expressions).
-
Multiply by the conjugate (square roots).
-
Use $\displaystyle\lim_{u\to 0}\frac{\sin u}{u}=1$ (sine with a matching denominator).
The same ideas let you work backwards. If a rational function has a removable discontinuity at $x=a$ with a given limit, the numerator must have the factor $(x-a)$.
Find the value of $k$ so that the function is continuous at $x = 4$:…
Find the value of $k$ so that the function is continuous at $x = 4$:
$f(x) = \begin{cases} \frac{\sqrt{x} - 2}{x - 4} & \text{if } x > 4 \\ kx + \frac{1}{4} & \text{if } x \le 4 \end{cases}$
Calculate the values of $a$ and $b$ so that the rational function $f(x) = \frac{x^2 + ax + b}{x^2 - 4}$ has a removable discontinuity at $x = 2$ and, furthermore, the…
Calculate the values of $a$ and $b$ so that the rational function $f(x) = \frac{x^2 + ax + b}{x^2 - 4}$ has a removable discontinuity at $x = 2$ and, furthermore, the limit of the function at that point is equal to $3$.
Study the continuity of the function $f(x)$ and determine the value of $k$ so that it is continuous at $x = 0$:…
Study the continuity of the function $f(x)$ and determine the value of $k$ so that it is continuous at $x = 0$:
$f(x) = \begin{cases} \frac{1 - \sqrt{1 - x^2}}{x^2} & \text{if } x < 0 \\ k & \text{if } x = 0 \\ \frac{\sin(x^2)}{2x^2} & \text{if } x > 0 \end{cases}$
Continuity in real-world models
Models built from different formulas on different time intervals are everywhere: heating and cooling, growth and saturation, track design. Continuity here has a physical meaning: the quantity cannot change instantly. Oven temperatures, populations, and heights of a roller coaster do not teleport.
The method is the same as before: match the two pieces at the joining time. Keep units and meaning in your answer.
An expression like $k\cdot 0.9^{t-10}$ with a base between $0$ and $1$ is exponential decay: the quantity shrinks by the same percent in equal time steps. That is how an oven cools after it is turned off.
The growth of a bacteria population in a laboratory culture is modeled by the function $P(t)$, where $t$ is the time in hours and $P(t)$ is the population in thousands…
The growth of a bacteria population in a laboratory culture is modeled by the function $P(t)$, where $t$ is the time in hours and $P(t)$ is the population in thousands of bacteria:
$P(t) = \begin{cases} 50(1 - e^{-0.2t}) & \text{if } 0 \le t < 10 \\ \frac{C}{t} & \text{if } t \ge 10 \end{cases}$
Calculate the value of the constant $C$ so that the population model is a continuous function at $t = 10$.
The temperature of an industrial oven in degrees Fahrenheit, $T(t)$, as a function of time in minutes $t$, is given by:…
The temperature of an industrial oven in degrees Fahrenheit, $T(t)$, as a function of time in minutes $t$, is given by:
$T(t) = \begin{cases} 20 + 15t & \text{if } 0 \le t < 10 \\ k \cdot 0.9^{t-10} & \text{if } t \ge 10 \end{cases}$
Determine the value of $k$ so that there is no sudden jump in temperature at 10 minutes. What physical phenomenon does the function represent from $t = 10$ onwards?
For the design of a roller coaster, two sections of track must be joined. The profile of the roller coaster is given by the following piecewise function, where $x$ is…
For the design of a roller coaster, two sections of track must be joined. The profile of the roller coaster is given by the following piecewise function, where $x$ is the horizontal distance in feet:
The equation of the profile is:
$f(x) = \begin{cases} \frac{100}{x + 10} + a & \text{if } 0 \le x < 10 \\ bx^2 - 2x + 15 & \text{if } x \ge 10 \end{cases}$
Knowing that the design requires the track to be continuous at the joining point $x = 10$, and that Section 1 starts from a height of $15\text{ ft}$ at $x = 0$, calculate the values of $a$ and $b$.
The Intermediate Value Theorem
If you walk from the bottom of a hill to the top without ever teleporting, you must pass through every elevation in between. That is the idea behind one of the most useful consequences of continuity.
To locate roots, make a table of values at consecutive integers and look for sign changes.
Equations of the form $f(x)=g(x)$. Move everything to one side and define $h(x)=f(x)-g(x)$. A solution of $f=g$ is a zero of $h$. This is how you prove that a company breaks even (revenue equals cost) at some moment.
Prove, using Bolzano's Theorem, that the equation $x^3 + 2x - 4 = 0$ has at least one real solution in the interval $(1, 2)$. Justify all the necessary conditions to…
Prove, using Bolzano's Theorem, that the equation $x^3 + 2x - 4 = 0$ has at least one real solution in the interval $(1, 2)$. Justify all the necessary conditions to apply the theorem.
The following graph illustrates the principle of Bolzano's Theorem for a continuous polynomial function $f(x) = x^3 - 4x - 2$. By observing the graph and evaluating the…
The following graph illustrates the principle of Bolzano's Theorem for a continuous polynomial function $f(x) = x^3 - 4x - 2$. By observing the graph and evaluating the function, find an interval of length 1 (that is, $[n, n+1]$ with $n \in \mathbb{Z}$) where the existence of a real root can be guaranteed, justifying your answer.
A technology company estimates that its revenue $I(x)$ and its costs $C(x)$, both in millions of dollars and as a function of the years $x$ of operation, are given by:…
A technology company estimates that its revenue $I(x)$ and its costs $C(x)$, both in millions of dollars and as a function of the years $x$ of operation, are given by:
$I(x) = 10 \ln(x + 1)$
$C(x) = e^{x - 2}$
Using Bolzano's Theorem, prove that there is an exact moment between the second and fifth year of operation ($x \in [2, 5]$) when the company reaches its break-even point (revenue equals costs).
Auxiliary functions and tricky endpoints
Many proofs ask you to show that two quantities become equal at some point, or that a function crosses a given curve. The strategy: build an auxiliary function whose zero is what you want, then apply the IVT.
The same recipe works when the data are real measurements. If a battery drops continuously from $100\%$ to $20\%$ over a trip, and you ask whether it ever equals some other changing quantity, subtract the two and check the signs at the start and end of the trip.
Sometimes the hypotheses give "$\le$" or "$\ge$" instead of strict inequalities. Then an endpoint might already be a zero, and you must handle that case separately.
Let $f$ be a continuous function on the closed interval $[0, 1]$ such that $0 \le f(x) \le 1$ for all $x \in [0, 1]$. Prove, using Bolzano's Theorem, that there is at…
Let $f$ be a continuous function on the closed interval $[0, 1]$ such that $0 \le f(x) \le 1$ for all $x \in [0, 1]$. Prove, using Bolzano's Theorem, that there is at least one point $c \in [0, 1]$ where the graph of the function intersects the parabola $y = x^2$, that is, prove that $f(c) = c^2$. (Hint: Define an auxiliary function $g(x) = f(x) - x^2$).
The battery charge level of an electric vehicle, $B(t)$ in percent, decreases continuously during a trip. At the initial instant $t = 0$ hours, the battery is at…
The battery charge level of an electric vehicle, $B(t)$ in percent, decreases continuously during a trip. At the initial instant $t = 0$ hours, the battery is at $100\%$. Upon arriving at the destination at $t = 5$ hours, the battery shows $20\%$. An engineer claims that, at some exact instant during the trip, the battery percentage coincided numerically with the expression $50 + 5t$.
Prove mathematically that the engineer is right using continuity theorems.
Back to the start: the hill is continuous (no jumps in elevation), while the parking fee $C(t)=2+1.5\lfloor t\rfloor$ has a jump discontinuity at every full hour. You can now test any function for that difference with limits, and you know that for continuous functions, the IVT guarantees you cannot skip over a value.