Theorems on continuity and differentiability
Continuity
We want to begin reminding continuity definition for real variabled functions:
Continuity definition
Lets f a real variabled function. It is said that f is continuous at \( x_{0} \) if
$$ \forall \epsilon>0, \exists \delta > 0 : |x – x_{0}| < \delta \Rightarrow |f(x) – f(x_{0}) | < \epsilon .$$
Or equivalently
$$ \lim_{x \mapsto x_{0}} f(x) = f(x_{0}) $$
Lets f a real variabled function. It is said that f is continuous at \( x_{0} \) if
$$ \forall \epsilon>0, \exists \delta > 0 : |x – x_{0}| < \delta \Rightarrow |f(x) – f(x_{0}) | < \epsilon .$$
Or equivalently
$$ \lim_{x \mapsto x_{0}} f(x) = f(x_{0}) $$
The first theorem we want to see is the Intermediate Value Theorem, witch states that all continuous functions takes all intermediate values between the image of extremes of interval.
Intermediate Value Theorem
Lets f a real variabled and continuous function with an interval [a, b]. If
$$ f(b) < f(a) \Rightarrow \forall y\in[f(a), f(b)], \exists x_{0} \in [a,b] : f(x_{0})=y.$$
Lets f a real variabled and continuous function with an interval [a, b]. If
$$ f(b) < f(a) \Rightarrow \forall y\in[f(a), f(b)], \exists x_{0} \in [a,b] : f(x_{0})=y.$$
Rolle’s Theorem
Lets f a real variabled and continuous and differentiable function with an interval [a, b]. If
$$ f(a) = f(b), \exists x_{0} \in [a,b] : f'(x_{0})=0.$$
Lets f a real variabled and continuous and differentiable function with an interval [a, b]. If
$$ f(a) = f(b), \exists x_{0} \in [a,b] : f'(x_{0})=0.$$
Following Theorem, is called Mean Value Theorem, implies differentiability. It states that exists at least a point wich derivate is equal to slope of secant rect passing through points (a, f(a)), (b, f(b)).
Mean Value Theorem
Lets f a real variabled and continuous and differentiable function with an interval [a, b] then
$$ \exists x_{0} \in [a,b] : f'(x_{0})=\frac{f(b)-f(a)}{b-a}.$$
Lets f a real variabled and continuous and differentiable function with an interval [a, b] then
$$ \exists x_{0} \in [a,b] : f'(x_{0})=\frac{f(b)-f(a)}{b-a}.$$
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