6.1 Definition of Continuity

Continuity is a fundamental concept in calculus, describing how functions behave without abrupt changes or breaks. It's crucial for understanding limits, derivatives, and integrals. This topic dives into the nitty-gritty of what makes a function continuous.

We'll explore the definition of continuity at a point and on intervals, learn how to test for continuity, and examine different types of discontinuities. These ideas form the backbone of more advanced calculus concepts you'll encounter later.

Continuity of functions

Definition of continuity at a point

• A function $f$ is continuous at a point $c$ if the following three conditions are satisfied:
  1. $f(c)$ is defined
  2. The limit of $f(x)$ as $x$ approaches $c$ exists
  3. The limit of $f(x)$ as $x$ approaches $c$ is equal to $f(c)$
• The three conditions for continuity at a point $c$ can be summarized as the function is defined at $c$, the limit exists at $c$, and the value of the function at $c$ is equal to the limit at $c$
• Example: Consider the function $f(x) = x^2$. To check if $f$ is continuous at $c = 1$, we evaluate $f(1) = 1^2 = 1$, confirm that the limit of $f(x)$ as $x$ approaches $1$ exists and equals $1$, and see that $f(1) = 1$ matches the limit value

Continuity on an interval

• A function is continuous on an interval if it is continuous at every point within that interval
• If a function is continuous on a closed interval $[a, b]$, then it is also continuous at the endpoints $a$ and $b$
  • Example: The function $f(x) = \sin(x)$ is continuous on the closed interval $[0, \pi]$ because it is continuous at every point within the interval, including the endpoints $0$ and $\pi$
• If a function is continuous on an open interval $(a, b)$, then it may or may not be continuous at the endpoints $a$ and $b$
  • Example: The function $f(x) = \frac{1}{x}$ is continuous on the open interval $(0, \infty)$ but is not continuous at the endpoint $0$ because the limit of $f(x)$ as $x$ approaches $0$ does not exist

Testing continuity at a point

Evaluating continuity using the definition

• To determine if a function is continuous at a point $c$, first check if the function is defined at $c$ by evaluating $f(c)$
• Next, evaluate the limit of $f(x)$ as $x$ approaches $c$ from both the left and right sides. If the left-hand and right-hand limits exist and are equal, then the limit of $f(x)$ as $x$ approaches $c$ exists
• Finally, compare the value of the limit (if it exists) to the value of $f(c)$. If they are equal, then the function is continuous at $c$
  • Example: Consider the function $f(x) = \frac{x^2 - 1}{x - 1}$. To check if $f$ is continuous at $c = 1$, we first note that $f(1)$ is undefined due to division by zero. However, the limit of $f(x)$ as $x$ approaches $1$ exists and equals $2$. Since $f(1)$ is undefined, the function is not continuous at $c = 1$
• If any of the three conditions for continuity are not satisfied, then the function is discontinuous at $c$

Identifying points of discontinuity

• To identify points of discontinuity, examine the function's definition and look for points where the function is undefined, has a jump in value, or has an infinite limit
  • Example: The function $f(x) = \frac{1}{x - 2}$ has a point of discontinuity at $x = 2$ because the function is undefined at this point due to division by zero
• Points of discontinuity can also be identified by examining the graph of a function and looking for gaps, jumps, or asymptotes
  • Example: The graph of the function $f(x) = \frac{1}{\sqrt{x}}$ has a point of discontinuity at $x = 0$ because the function is undefined for negative values of $x$, resulting in a gap in the graph at $x = 0$

Types of discontinuities

Removable discontinuities

• A removable discontinuity occurs when a function is undefined at a point $c$, but the limit of the function as $x$ approaches $c$ exists
• The function can be made continuous by redefining the value at the point of discontinuity to match the limit
  • Example: The function $f(x) = \frac{x^2 - 1}{x - 1}$ has a removable discontinuity at $x = 1$. By redefining $f(1) = 2$ to match the limit value, the function becomes continuous at $x = 1$

Jump discontinuities

• A jump discontinuity occurs when a function has a defined value at a point $c$, but the left-hand and right-hand limits as $x$ approaches $c$ exist and are not equal
• The function "jumps" from one value to another at the point of discontinuity
  • Example: The function $f(x) = \begin{cases} 1, & x < 0 \\ 2, & x \geq 0 \end{cases}$ has a jump discontinuity at $x = 0$ because the left-hand limit is $1$ and the right-hand limit is $2$

Infinite discontinuities

• An infinite discontinuity occurs when the limit of the function as $x$ approaches $c$ from either the left or right side (or both) is infinite
• This can happen when the function approaches positive or negative infinity as $x$ approaches $c$
  • Example: The function $f(x) = \frac{1}{x}$ has an infinite discontinuity at $x = 0$ because the limit of $f(x)$ as $x$ approaches $0$ from both the left and right sides is infinite

Other types of discontinuities

• Oscillating discontinuities occur when the function oscillates rapidly near the point of discontinuity
  • Example: The function $f(x) = \sin\left(\frac{1}{x}\right)$ has an oscillating discontinuity at $x = 0$ because the function oscillates more and more rapidly as $x$ approaches $0$
• Mixed discontinuities exhibit a combination of the above types
  • Example: The function $f(x) = \begin{cases} \frac{1}{x}, & x < 0 \\ 2, & x = 0 \\ x^2, & x > 0 \end{cases}$ has a mixed discontinuity at $x = 0$, with an infinite discontinuity from the left and a jump discontinuity from the right