define what it means for f(x) to be continuous at the point x=a. what does it mean if f(x) is continuous on the interval [a,b]? what does it mean to say f(x) is

Question

define what it means for f(x) to be continuous at the point x=a. what does it mean if f(x) is continuous on the interval [a,b]? what does it mean to say f(x) is continuous?

Answer 1

f(x) is continuous at a if...

f(a) exists (is a Real number, C) AND

lim as x→a of f(x) = C.

Answer 2

For the 1st part,

 

lim f(x) = f(a) = lim  f(x)

x→a^-                  x→a⁺

 

 

For the 2nd part,

 

f(x) is only differentiable between x=a and x=b.  This means that the derivative is not undefined anywhere within this interval.

 

 

For the 3rd part,

 

f(x) is differential in the open interval (-∞, ∞).  The derivative is defined at all points.