Calculus Therorems and Continuity

K

________ is any number between f (a) and f (b)

Jump Discontinuity

________: limits from either side are not equal.

Infinite Discontinuity

________: occurs at vertical asymptotes.

Removable Discontinuity

________: limits from either side agree, but are not equal to f (x), denominator goes to 0, but if factored, one factor can cancel.

Jump Discontinuity

limits from either side are not equal

Removable Discontinuity

limits from either side agree but are not equal to f(x); denominator goes to 0, but if factored, one factor can cancel

Infinite Discontinuity

occurs at vertical asymptotes

Squeeze Theorem

If g(x) ≤ f(x) ≤ h(x)

and

lim x→c g(x) = lim x→c h(x) = L

then

lim x→c f(x) = L

Continuity

A function is continous at a point (c) if the limit at that point exists and is equal to the value of the function at that point
limx->c f(x) = f(c)

Intermediate Value Theorem

If x is on the closed interval [a,b] and

f(a) ≠ f(b) and

k is any number between f(a) and f(b)

then there exists some number (c) in [a,b] such that f(c) = k

Mean Value Theorem

if f is continuous on the closed interval \[a,b\] and differentiable on the open interval (a,b), then there exists at least one number, c, within (a,b) in which

f’(c) = f(b)-f(a)/b-a

Extreme Value Theorem

if f is continuous on the closed interval \[a,b\], then f has both a global minimum and a global maximum