$Proving f is continuous$

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$Proving f is continuous$

4. Consider the function f : R → R, f(x) = ⌊x⌋, given by discarding the decimal
part of x: f(x) = max {y ∈ Z | y ≤ x}. For example, ⌊5.8⌋ = 5 and ⌊−2.3⌋ =
−3. Find all points at which f is not continuous and prove this fact. Prove that
f is continuous at all other points.

Calculus

Stickix02

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We need to check continuity of the function $$f(x)=\lfloor x\rfloor$$
Case I: Let us select any random real number $a$ such that $a\notin Z$ where $Z$ is integer. Then
$$\lim_{x\to a^-}\lfloor x\rfloor=\lim_{x\to a^+}\lfloor x\rfloor=a=f(a)$$
continuity condition satisfied, hence it is continuous at all non-integer points.
Case II: Let us select any random real number $a$ such that $a\in Z$ where $Z$ is integer. Then left hand limit $$\lim_{x\to a^-}\lfloor x\rfloor=\lfloor a^-\rfloor=a-1$$ and right hand limit $$\lim_{x\to a^+}\lfloor x\rfloor=\lfloor a^+\rfloor=a$$
So, continuity condition is not satisfied, hence it is discontinuous at all integer points.

Aman R

642

$Proving f is continuous$

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