continuous functions

Question 1

interval:

Answer 1

a nonempty subset of the form (a,b), [a,b], etc.

Question 2

continuity (formal definition):

Answer 2

let S in the reals and f:S->R be a function. pick a point x0 in S. f is continuous at the point x0 if ∀ε > 0, ∃δ > 0, ∀x ∈ S, such that (|x − x0| < δ) ⇒ (|f (x) − f (x0)| < ε)
the function is continuous if that is true for all x0 in S

Question 3

for a point x0 in S, the following are equivalent:

Answer 3

f is continuous at x0
for every (monotone) sequence (an)n in S, if (an)n converges to x0, then the sequence (f(an))n converges to f(x0)

Question 4

if f and g are continuous at x0:

Answer 4

f+g is continuous at x0
fg is continuous at x0
f/g is continuous at x0 if g(x)!=0 for all x

Question 5

if f:S->r and g:T->R with f(S) in T:

Answer 5

if f and g are continuous at x0 and f(x0) respectively, g.f:S->R is continuous at x0

Question 6

boundedness theorem:

Answer 6

if f:[a,b]->R is continuous, f is bounded

Question 7

intermediate value theorem:

Answer 7

let f:[a,b]->R be continuous. then, every number between a and b is attained by f (i.e. for any c and d in [a,b] there is some f(d)=c