📚 Topic Summary
In calculus, continuity describes functions without abrupt breaks or jumps. A function $f(x)$ is continuous at a point $x = c$ if three conditions are met: 1) $f(c)$ is defined, 2) $\lim_{x \to c} f(x)$ exists, and 3) $\lim_{x \to c} f(x) = f(c)$. The formal definition uses epsilon-delta notation to rigorously define the limit. This quiz helps you understand and apply these concepts!
🧠 Part A: Vocabulary
Match the term with its correct definition:
- Term: Limit
- Term: Continuity
- Term: Epsilon ($\epsilon$)
- Term: Delta ($\delta$)
- Term: Function
- Definition: A relation where each input has only one output.
- Definition: The value that a function "approaches" as the input approaches some value.
- Definition: A function is continuous at a point if the limit exists at that point, the function is defined at that point, and the limit equals the function's value at that point.
- Definition: A small positive number representing the maximum allowed error in the function's output.
- Definition: A small positive number representing the maximum allowed difference in the function's input from a specific point.
(Match each term 1-5 with its correct definition 1-5)
✍️ Part B: Fill in the Blanks
A function $f(x)$ is said to be _______ at a point $x = c$ if, for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - c| < \delta$, then $|f(x) - f(c)| < \epsilon$. This means that we can make the function's output arbitrarily close to $f(c)$ by choosing $x$ sufficiently _______ to $c$. In simpler terms, the graph of the function has no _______ at $x = c$. This is the _______ definition of continuity.
🤔 Part C: Critical Thinking
Explain, in your own words, why all three conditions (f(c) is defined, the limit exists at c, and the limit equals f(c)) are necessary for a function to be continuous at a point. Give an example of a function that fails one of these conditions and is therefore discontinuous.