General Math I - Lecture 7: Continuity of Functions

Which of the following functions is continuous at x=0?

At which point is the function g(x) not continuous?

What is the value of f(0) if f(x) = e for x < 0 and f(x) = -x for x ≥ 0?

Which statement regarding continuity at x=-3 for the function f(x) is true?

Which of the following functions is defined but not continuous at its specified point?

For the function k(x) described, where is the function continuous?

Which of the following is a requirement for continuity at a point?

Which function is continuous everywhere except at x=-3?

General Math (I) - Lecture 7

Topic: Continuity of Functions
Chapter: 2 - Functions & Limits
Key Concepts: Continuity, Limits, Functions
Study points
Continuity at a point
A function f is continuous at x = a when:
f(a) is defined
lim (x→a) f(x) exists
lim (x→a) f(x) = f(a)
If any condition is not met, the function is not continuous at x = a.
Continuity on domains
Constant functions
Polynomial functions
Exponential functions
Absolute value functions
Odd root functions (e.g., cube root)
Sine and cosine functions
Continuity characteristics
Sums (f + g)
Differences (f - g)
Products (f ⋅ g)
Constant multiples (k ⋅ f, where k is any number)
Quotients (f/g, provided g(c) ≠ 0)
Powers (fr/s, provided it is defined on an open interval containing c, where r and s are integers)
Continuity on an interval
A function is continuous on a closed interval if it is continuous at all interior points and the endpoints.
A function f is continuous on the closed interval [a, b] if:
f is defined on [a, b]
f is continuous on (a, b)
lim (x→a+) f(x) = f(a)
lim (x→b-) f(x) = f(b)

Explore the concept of continuity in functions in this lecture quiz. Delve into the conditions that define continuity at a point and learn about various types of functions, including polynomial and exponential. This quiz will test your understanding of limits and characteristics of continuous functions.