Continuity of a function at a point
In 1st Bachillerato you checked the continuity of polynomials, simple rational functions and piecewise functions made of lines or parabolas. In 2nd Bachillerato the same three conditions apply to more elaborate functions: absolute values, exponentials, logarithms and trigonometric functions combined in a single piece.
The three conditions, once again
Check whether the function $f(x) = e^{x-1}$ is continuous at the point $x = 1$. For a complete answer, you must verify whether the value of the function at that point…
Check whether the function $f(x) = e^{x-1}$ is continuous at the point $x = 1$. For a complete answer, you must verify whether the value of the function at that point coincides with the limit of the function as $x$ approaches 1.
Study the continuity of the function $f(x) = \frac{x^2 - 1}{x - 1}$ at the point $x = 1$. If it has a discontinuity, state what type it is (removable, jump, or infinite)…
Study the continuity of the function $f(x) = \frac{x^2 - 1}{x - 1}$ at the point $x = 1$. If it has a discontinuity, state what type it is (removable, jump, or infinite) and explain whether there is any way to redefine the function so that it is continuous at that point.
Discontinuities with absolute values
The absolute value changes expression at the point where its argument is zero, just like a piecewise function. To study continuity there, you have to remove the absolute value piece by piece and compare the one-sided limits.
Study the continuity of the following function across its domain, classifying any points of discontinuity you find: $$f(x) = \frac{|x - 2|}{x^2 - x - 2}$$ Accompany your…
Study the continuity of the following function across its domain, classifying any points of discontinuity you find:
$$f(x) = \frac{|x - 2|}{x^2 - x - 2}$$Accompany your analytical study with a diagram of the one-sided limits at the critical points.
Given the function $f(x) = \frac{1}{x + 4}$, justify why it is not continuous at the point $x = -4$. To do so, calculate the one-sided limits $\lim_{x \to -4^-} f(x)$…
Given the function $f(x) = \frac{1}{x + 4}$, justify why it is not continuous at the point $x = -4$. To do so, calculate the one-sided limits $\lim_{x \to -4^-} f(x)$ and $\lim_{x \to -4^+} f(x)$ and describe the type of discontinuity it has.
Discontinuities with exponential, logarithmic and trigonometric functions
When the denominator or the exponent of a function grows without bound, the one-sided limits sometimes give different results as $x\to a^-$ and as $x\to a^+$: this is where jump discontinuities appear, even though the function looks well behaved.
Study the continuity of the function $f(x) = \frac{1}{1 + 2^{1/x}}$ at the point $x = 0$. To do so, calculate the one-sided limits $\lim_{x \to 0^-} f(x)$ and…
Study the continuity of the function $f(x) = \frac{1}{1 + 2^{1/x}}$ at the point $x = 0$. To do so, calculate the one-sided limits $\lim_{x \to 0^-} f(x)$ and $\lim_{x \to 0^+} f(x)$. What type of discontinuity does the function have at that point?
An industrial cooling system regulates its power $P$ (in kilowatts) according to the outside temperature $T$ (in degrees Celsius) by means of the following function:…
An industrial cooling system regulates its power $P$ (in kilowatts) according to the outside temperature $T$ (in degrees Celsius) by means of the following function:
$$P(T) = \begin{cases} \dfrac{a \cdot e^T}{e^T + 1} & \text{if } T \leq 0 \\[2mm] \dfrac{\sqrt{T+4} - 2}{T} & \text{if } T > 0 \end{cases}$$Determine the value of the constant $a$ so that the power transition is continuous when the temperature passes through $0\,^\circ C$.
Piecewise functions: finding the parameters for continuity
The most typical exam question (EvAU): a piecewise function with one or two unknown letters, and you have to find the values that avoid any jump at the points where the pieces join.
Given the function dependent on the parameter $k \in \mathbb{R}$: $$f(x) = \begin{cases} \frac{x^2 - kx}{x - 3} & \text{if } x < 3 \\ 2x - 5 & \text{if } x \geq 3…
Given the function dependent on the parameter $k \in \mathbb{R}$:
$$f(x) = \begin{cases} \frac{x^2 - kx}{x - 3} & \text{if } x < 3 \\ 2x - 5 & \text{if } x \geq 3 \end{cases}$$Calculate the value of $k$ so that the function has a jump discontinuity (finite jump) at $x = 3$. Is there any value of $k$ that makes it continuous at that point? Justify your answer by computing the one-sided limits.
A team of robotics engineers has programmed the motion of a mechanical arm. The speed of the end of the arm (in m/s) follows the path:…
A team of robotics engineers has programmed the motion of a mechanical arm. The speed of the end of the arm (in m/s) follows the path:
$v(t) = \begin{cases} \frac{\sqrt{t+5}-3}{t-4} & \text{if } 0 \leq t < 4 \\ \frac{a}{12} & \text{if } t = 4 \\ \frac{t^2 - 16}{t^2 - 3t - 4} \cdot b & \text{if } t > 4 \end{cases}$
Determine the values of $a$ and $b$ so that the robot's motion is continuous (without sudden jerks) at time $t = 4$ seconds.
In the design of a new component for a particle accelerator, the magnetic field strength as a function of position $x$ (in millimetres) is defined by the following…
In the design of a new component for a particle accelerator, the magnetic field strength as a function of position $x$ (in millimetres) is defined by the following function:
$f(x) = \begin{cases} \frac{e^{ax} - 1}{x} & \text{if } x < 0 \\ b & \text{if } x = 0 \\ \frac{x^2 - 4}{x - 2} \cdot \cos(\pi x) & \text{if } x > 0 \end{cases}$
Determine the values of the real parameters $a$ and $b$ so that the magnetic field strength is continuous at the point $x = 0$. Once found, would the function be continuous at the point $x = 2$? Reason your answer analytically.