3.1 Definition and types of continuity

Continuity is a crucial concept in calculus, describing how functions behave without breaks or jumps. It's the foundation for understanding limits and derivatives, helping us analyze smooth, predictable function behavior.

Continuous functions are like unbroken pencil lines – no lifting the pencil off the paper. We'll explore different types of discontinuities, learn how to spot them, and see why continuity matters in real-world applications.

Continuity

Continuity at points and intervals

• Function $f(x)$ is continuous at point $a$ if:
  • $f(a)$ exists (the function is defined at point $a$)
  • $\lim_{x \to a} f(x)$ exists (the limit of the function as $x$ approaches $a$ exists)
  • $\lim_{x \to a} f(x) = f(a)$ (the limit of the function as $x$ approaches $a$ equals the function value at $a$)
• Function is continuous on interval $[a, b]$ if it is continuous at every point within the interval
  • No breaks, holes, or jumps in the graph of the function on the interval ($\sin(x)$ on $[-\pi, \pi]$)
  • Function is defined at every point in the interval (polynomial functions)

Types of function discontinuities

• Removable discontinuity occurs when:
  • $\lim_{x \to a} f(x)$ exists, but $f(a)$ is either undefined or not equal to the limit value
  • Appears as a hole in the graph of the function ($\frac{x^2-1}{x-1}$ at $x=1$)
• Jump discontinuity happens when:
  • Both one-sided limits, $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$, exist but have different values
  • Graph shows a jump at the point of discontinuity (step functions, $\lfloor x \rfloor$ at integer values)
• Infinite discontinuity arises when:
  • At least one of the one-sided limits, $\lim_{x \to a^-} f(x)$ or $\lim_{x \to a^+} f(x)$, is infinite
  • Graph has a vertical asymptote at the point of discontinuity ($\frac{1}{x}$ at $x=0$)

Determining function continuity

• Using the definition:

1. Check if $f(a)$ is defined at the point of interest
2. Evaluate $\lim_{x \to a} f(x)$ using limit laws and techniques
3. Compare the values of $\lim_{x \to a} f(x)$ and $f(a)$; if equal, the function is continuous at $a$

• Examining the graph:
  • Identify any holes (removable discontinuities), jumps (jump discontinuities), or vertical asymptotes (infinite discontinuities)
  • If the graph is unbroken with no holes, jumps, or vertical asymptotes, the function is continuous on its domain (trigonometric functions on their respective periods)

Applications of continuous functions

• Properties of continuous functions:
  • Sum, difference, product, and quotient of continuous functions are also continuous ($f(x)=x^2+\sin(x)$ is continuous on $\mathbb{R}$)
  • Composition of continuous functions is continuous ($f(x)=\cos(\ln(x))$ is continuous on $(0,\infty)$)
  • Intermediate Value Theorem: If $f(x)$ is continuous on $[a, b]$ and $k$ is between $f(a)$ and $f(b)$, then there exists a $c \in [a, b]$ such that $f(c) = k$
• Solving problems using continuity:
  • Determine the continuity of combined or composed functions using the properties of continuous functions
  • Apply the Intermediate Value Theorem to prove the existence of solutions to equations ($\cos(x)=x$ has a solution on $[0, 1]$) or intersections of graphs ($y=x$ and $y=\cos(x)$ intersect on $[0, 1]$)