┄
Continuous at c
｢f(x) Ƚx￫c｣ = f(c)
┄
A function f:A￫ℝ is differentiable at c if and only if
｢(f(x)−f(c))/(x−c) Ƚx￫c｣
exists.
The value of the limit, called the derivative of f at c, is denoted by f(x)ᐁx｢c｣.
┄
Differentiable at c
｢(f(x)−f(c))/(x−c) Ƚx￫c｣ = f(x)ᐁx｢c｣
┄
If f is differentiable at c then f is continuous at c.
If f is Riemann integrable on [a,b] then f(t)ᐃt｢a,x｣ is continuous on [a,b]. Further more, f is continuous on [a,b] then f(t)ᐃt｢a,x｣ is differentiable on [a,b] and (f(t)ᐃt｢a,x｣)ᐁx=f.
┄
Integrable on [a,b]
(f(t)ᐃt｢a,x｣)ᐁx = f
┄
ᐥf is continuous ⇒
❶ f is integrable
❷ f(x)ᐃx is differentiable
❸ (f(x)ᐃx)ᐁx = fᐥ
