Continuity and Discontinuities
Master the formal definition of continuity, types of discontinuities, and the Intermediate Value Theorem.
Keyboard Shortcuts
💡 Pro tip: Use keyboard shortcuts for faster studying!
Study Smart Tips for Continuity and Discontinuities
Master these concepts using proven study techniques that actually work:
Active Recall
Test yourself before flipping each card to strengthen memory retention
Spaced Repetition
Review difficult cards more frequently than easy ones
Multiple Sessions
Break study time into shorter, focused sessions
Explain Aloud
Verbalize answers to reinforce understanding
Questions Covered in This Set
8 cards to master
What are the three conditions for a function f(x) to be continuous at x = a?
1) f(a) exists, 2) lim(x→a) f(x) exists, 3) lim(x→a) f(x) = f(a)
What is a removable discontinuity?
A discontinuity where the limit exists but either f(a) is undefined or f(a) ≠ lim(x→a) f(x). It can be 'fixed' by redefining the function at that point.
What is a jump discontinuity?
A discontinuity where the left-hand and right-hand limits exist but are not equal, causing the function to 'jump' from one value to another.
What is an infinite discontinuity?
A discontinuity where the function approaches ±∞ as x approaches a, typically occurring at vertical asymptotes.
State the Intermediate Value Theorem.
If f is continuous on [a,b] and N is any number between f(a) and f(b), then there exists at least one c in (a,b) such that f(c) = N.
Why does f(x) = (x² - 4)/(x - 2) have a discontinuity at x = 2?
Because f(2) is undefined (denominator equals zero), even though the limit as x→2 exists and equals 4.
Which types of functions are continuous everywhere on their domain?
Polynomials, exponential functions, logarithmic functions, trigonometric functions, and compositions/combinations of continuous functions.
How can the IVT be used to guarantee the existence of roots?
If a continuous function changes sign over an interval (f(a) and f(b) have opposite signs), the IVT guarantees the function crosses the x-axis, meaning a root exists in that interval.
