Maths continuity

Bolzano’s theorem

If f(x) is continuous in [a,b], f(a)<0 and f(b)>0 or viceversa. Then exists the value x=0 in the open interval (a,b) that f(c)= 0

Theorem of the fixed point

If f(x) is continuous in [a,b], then a point exists in [a,b] in which f(p)=p

Daboux theorem

If f(x) is continuous in [a,b] and f(a)<k and f(b)>k or vice versa for some k∈ Im (f). Then exists c∈(a,b) that f(c)= k

Weierstrass theorem

If f(x) is continuous in [a,b] then there will be a global maximum and minimum in [a,b]