Continuity of a Function, Definition, 3 Conditions, Discontinuities, Practice Examples - Calculus

In calculus, a function is continuous if its graph can be drawn without lifting your pen, meaning there are no breaks or jumps. More formally, a function f(x) is continuous at a point a if three conditions are met: f(a) is defined, the limit of f(x) as x approaches a exists, and the limit equals f(a).

💡Intuitive Understanding
• No Gaps or Jumps: You can draw the graph of a continuous function without your pencil leaving the paper.
• Small Input, Small Output: A small change in the input (x) results in a small change in the output (f(x)).

💡The Formal Definition (at a Point a)
For a function f(x) to be continuous at a point a, all three of the following conditions must be true:
• f(a) is defined: The function must have a specific, defined value at the point a.
• The limit of f(x) as x approaches a exists: The function must approach the same value as x gets closer and closer to a from both the left and right sides.
• The limit equals the function value: The value the function approaches (the limit) must be the same as the actual value of the function at a, i.e., lim x→a f(x) = f(a).

💡Discontinuities
If any of these three conditions are not met, the function is discontinuous at that point. Common types of discontinuities include:
• Removable: A "hole" in the graph that could be "filled".
• Jump: The graph jumps from one value to another, like the function f(x) = {-1 for x less than or equal to 0 and 1 for x greater than 0 at x=0}.
• Infinite: A vertical asymptote where the function's value goes to positive or negative infinity.

💡Worksheets are provided in PDF format to further improve your understanding:
- Questions Worksheet: https://drive.google.com/file/d/1IfdCaeJTszq4Is48tDhUUxlMd-w9Eltx/view?usp=drive_link
- Answers: https://drive.google.com/file/d/11PKq7Z-aEJQOqR_xbqFdAEQdQH4JjKnu/view?usp=drive_link

💡Chapters:
00:00 Types of discontinuities
01:58 Discontinuities, examples
04:26 Continuity at a point, with examples
08:16 Continuity over an interval, with example
10:55 Removing discontinuities, with example
13:42 Infinite limits and vertical asymptotes
14:41 Infinite limits and horizontal asymptotes
15:34 Relative magnitudes of functions

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