f(x) = \left\{\begin{matrix} 1 & x \epsilon Q \\ 0 & x \epsilon \frac{\mathbb{R}}{Q}...


{eq}f(x) = \left\{\begin{matrix} 1 & x \epsilon Q \\ 0 & x \epsilon \frac{\mathbb{R}}{Q} \end{matrix}\right.{/eq} Dirichlet function

f is not continuous at any point of {eq}\mathbb{R}{/eq}. Take {eq}C \epsilon Q{/eq} and {eq}(x_n) \epsilon \frac{\mathbb{R}}{Q}{/eq} such that {eq}x_n \rightarrow C{/eq} (How we can do that?) Find a unique sequence for {eq}f(x_n) = 0{/eq} for all {eq}n \epsilon N{/eq}

Alternative definitions of limits:

Besides the usual {eq}\epsilon-\delta{/eq} definition of limit, there is another one that can be stated in terms of convergent sequences:

Theorem: Let {eq}f:\mathbb{R}\to \mathbb{R}{/eq} be a real function, then {eq}\lim_{x\to C} f(x) = L {/eq} if and only if, for every sequence {eq}(c_n)_{n\in \mathbb{N}}{/eq} that converges to {eq}C{/eq}, we have that {eq}(f(c_n))_{n\in \mathbb{N}}{/eq} converges to {eq}L{/eq}.

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We are going to prove that {eq}f{/eq} is not continuous at any point {eq}C{/eq}, for this, it suffices to prove that {eq}\lim_{x\to C} f(x){/eq} limit...

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Continuous Functions Theorems


Chapter 5 / Lesson 10

Continuous functions are functions that have their conditions satisfied between multiple points, appearing as an uninterrupted line when graphed. See examples of how this is represented in the intermediate value theorem and the extreme value theorem.

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