📚 Topic Summary
Continuity on an interval means a function is continuous at every point within that interval. 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) $\lim_{x \to c} f(x) = f(c)$. When dealing with closed intervals $[a, b]$, we also need to check for one-sided continuity: continuity from the right at $a$ and continuity from the left at $b$.
In simpler terms, you can draw the function without lifting your pencil within the specified interval. This concept is fundamental for understanding more advanced topics in calculus.
🧠 Part A: Vocabulary
Match the term with its correct definition:
| Term |
Definition |
| 1. Continuous Function |
A. A function that is continuous from the left at that point. |
| 2. Interval |
B. A function where there are no breaks or jumps in its graph. |
| 3. Limit |
C. The value that a function approaches as the input approaches some value. |
| 4. Continuity from the Right |
D. A set of real numbers between two specified values. |
| 5. Continuity from the Left |
E. A function that is continuous from the right at that point. |
✍️ Part B: Fill in the Blanks
A function is said to be __________ on an interval if it is continuous at every point in the interval. At an endpoint 'a', the function must be continuous from the __________, and at an endpoint 'b', it must be continuous from the __________. A __________ is the y-value that the function approaches as x gets closer and closer to a particular value. For a function to be continuous at a point $c$, $f(c)$ must be __________, the $\lim_{x \to c} f(x)$ must __________, and these two values must be __________.
🤔 Part C: Critical Thinking
Explain, in your own words, why understanding continuity on intervals is important for studying calculus. Give a specific example of a calculus concept that relies on continuity.