need help...analysis related...

[Semidevil]

been thinking about this for hours, but dont know how to connect.

A funtion is continuous iff for all sequence X(n) that converges to c, the sequence f(X(n)) converges to f(c).

ok, so I break it into 2 parts ----->, and <---------

first part is If X(n) converges to c, then the seququence f(X(n)) convergs to f(c).

ok, so by definition, suppose X(n) converges to c. This means that the limit of X(n) is c. and that is where I am stuck. how do I jump from this, to f(X(n)) converges to f(c).

so if I try part 2, im still stuck. if f(x(n)) converges to f(c), then X(n) converges to c.

suppose f(X(n)) converges to f(c). This means the limit of f(x(n)) is f(c), which also means that it is continuous....now how do I make that jump to X(n) converges to c??

 

[chuckywang]

Damn...I don't know much about analysis, but it sounds as if you have to use the definition of the limit. i.e. lim X(n) as n->infinity = c means X(n*) - c > epsilon for sufficiently large n*, where epsilon is any positive real.

 

[desteffy]

show by contrapositive, assume the function is NOT continuous at a point c, and then this implies there exists x_n-->c where f(x_n) will not converge to f(c)
then you are done