Lecture #4: Continuity, Squeeze Theorem, and IVT (21-120)

Vocabulary flashcards covering Squeeze Theorem, continuity, discontinuities, and IVT from Lecture #4.

Squeeze Theorem

If f(x) ≤ g(x) ≤ h(x) near a (x ≠ a) and lim x→a f(x) = lim x→a h(x) = L, then lim x→a g(x) = L.

Upper bound

A function that lies at or above another function g(x) near a, used to bound g from above in the Squeeze Theorem.

Lower bound

A function that lies at or below another function g(x) near a, used to bound g from below in the Squeeze Theorem.

Bounded by two functions near a

Describes g(x) being trapped between two functions f and h whose limits coincide.

Limit as x approaches a

The value that f(x) or g(x) approaches as x gets arbitrarily close to a.

Continuity at a

A function f is continuous at a if f(a) is defined, lim x→a f(x) exists, and lim x→a f(x) = f(a).

Continuous on an interval

A function that is continuous at every point in that interval.

Jump discontinuity

A discontinuity where the left-hand and right-hand limits exist but are not equal.

Infinite (essential) discontinuity

A discontinuity where a limit from at least one side does not exist or is infinite.

Removable discontinuity

A discontinuity where the limit exists but the function value at that point does not equal the limit.

Continuous function families

Polynomials, rational functions, root functions, trig functions, inverse trig functions, exponential functions, and logarithmic functions are continuous at every number in their domain.

Intermediate Value Theorem (IVT)

If f is continuous on [a,b] and f(a) ≠ f(b), then every value between f(a) and f(b) is attained by f on [a,b].

Continuity from the right

f is continuous from the right at a if f(a) exists, lim x→a+ f(x) exists, and lim x→a+ f(x) = f(a).

Continuity from the left

f is continuous from the left at b if f(b) exists, lim x→b− f(x) exists, and lim x→b− f(x) = f(b).