Continuity

The definition of continuity

Continuity theorems

The intermediate value theorem

Continuity and the limit laws together

Which trio of conditions together defines continuity of f at the point a?

The function g equals (x^2 - 1)/(x - 1) for x not 1 and is left undefined at x = 1. What kind of discontinuity does g have at x = 1, and can it be repaired?

The intermediate value theorem promises a solution of f(c) = N between a and b. Which hypothesis is essential?

Which computation correctly evaluates lim x to 0 of e^(sin x)?

The function f(x) = x^2 for x at most 1 and f(x) = 2x - 1 for x greater than 1 is tested at the join x = 1. Which statement is correct?

A function is continuous on the closed interval [0, 3]. Which guarantee does the extreme value theorem provide?

Every polynomial is continuous on the whole real line. How many discontinuities does p(x) = x^2 + 1 have? Answer with a number.

f(x) = x^3 - x - 1 satisfies f(1) = -1 and f(2) = 5. By the intermediate value theorem, f attains the value 0 somewhere in (1, 2). What value does the theorem guarantee is attained? Answer with a number.

The function (x^2 - 1)/(x - 1) has limit 2 at x = 1. Defining g(1) equal to that limit removes the discontinuity. What value should g(1) be? Answer with a number.

A step function has left limit 2 and right limit 5 at x = 1. What is the size of the jump, the right limit minus the left limit? Answer with a number.