MHB How Does Continuity of T(n) for n≤2 Impact Asymptotic Bounds?

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Continuity of T(n) for n≤2 is relevant when determining asymptotic bounds for recurrence relations. It suggests that T(n) may behave predictably in that range, potentially simplifying analysis. However, the Master Theorem does not typically account for this continuity in its application. Clarification on whether "continuous" was intended to mean "constant" for n≤2 is also discussed. Understanding these nuances is crucial for accurate asymptotic analysis.
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Hello! (Wave)I am given some recurrence relations $T(n)$ and I have to give asymptotic upper and lower bounds for $T(n)$.
We assume that $T(n)$ is continuous for $n \leq 2$.
How can we use the fact that $T(n)$ is continuous for $n \leq 2$? (Thinking)
 
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Probably meant "constant for $n \leq 2$".
 
Bacterius said:
Probably meant "constant for $n \leq 2$".

Nice, thank you! (Smile)

We don't take this fact into consideration when we use the Master Theorem, right?
 

Thread 'Formal derivation of statement from Peano Arithmetic system'

I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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