THE MOST IMPORTANT THEOREMS OF ANALYSIS

LIMITS

Limit Uniqueness

If for x that goes to x0 the function f(x) has for limit the real number l, than this limit is unique.

Sign Permanence

If the limit of a function for x that goes to x0 is a number l different from 0, surely exists an open interval I of x0 (excluded at most x0) in which f(x) and l are both positive or both negative.

Sandwich Theorem

Are h(x), f(x) and g(x) three functions defined in the same domain D subset of R, excluded at most a point x0. If in each point different to x0 of the domain it is h(x)≤f(x)≤g(x), and the limit of the two functions h(x) and g(x), for x that goes to x0, is a same number l, than the limit of f(x) too for x that goes to x0 is equal l.

CONTINUITY

Weierstrass Theorem

If f(x) is a continuous function in a limited and close interval [a;b],
therefore this function has, in this interval, a global maximum and
a global minimum.

Intermediate Values Theorem

If f(x) is a continuous function in a limited and close interval [a;b],
therefore this function has at least one time all the values
between the maximum and the minimum.

Intermediate Zero Theorem

If f(x) is continuous in a closed interval [a;b] and f(a) and f(b) have opposite signs, there is a point in which f(x) is equal to 0.

DIFFERENTIABLE

Rolle

...and f(a)= f(b), then exist at least a points in which the first derivative is equals to zero.

Lagrange

De L'Hospital

Fermat

Cauchy

...then exist at least a point where [f(b)-f(a)] / [f(b)-
f(a)]=f'(c).

...and g(x) is continuous, defined and differentiable in
the same interval, and g'(x) is never equal to zero, then [f(b)-
f(a)]/[g(b)-g(a)]=f'(x)/g('(x)
in at least one point.

...limit for x that tends to xo of f(x)/g(x) equals to the
limit for x that tends to xo of the ratio of their first derivatives.

...and it has a mimimum or a maximum in the interval, in
this point f'(x)=0.

If a function that is defined on a closed and
limited interval [a;b] ,f(x) is continuous,
differentiable in the same interval...