Is this piecewise function differentiable?

[cpsantos]

Hi there,

I have a question about the piecise function A'. Is A' a differentiable function everywhere? Is it also differentiable in the junction points?

A' = A₁/((1+e^b(m-O_m))(1+e^bn)) + A₂/(1+e^b(m-O_m)) + A₃/((1+e^b(m-O_m))(1+e^bn))

where A₁, A₂ and A₃ are as defined below and b = 500. The value of A' alternates between these three different values, A₁, A₂ and A₃, depending on the current values of m and n variables. The m and n variables vary continusously and sinusoidally.


For 0 < t <= t₁

A₁ = D(t)/[((pi/2 - 1 + sin(w₁ t)) / w₁) + ((pi + 2) / w₂) + ((pi - 1) / w₃) - t]

For t₁ < t <= t₂

A₂ = D(t)/[((pi/2) / w₁) + ((pi + 1 + cos(w₂ (t - T₁)) / w₂) + ((pi/2 - 1) / w₃) - t]

For t₂ < t <= t₃

A₂ = D(t)/[((pi/2) / w₁) + ((pi/2) / w₂) + ((pi/2 - cos(w₃ (t - T_{1 - T2})) / w₃) - t]

A₃ = D(t)/[(pi/2-1+sin(w₁ t)/w₁)+ (pi+2/w₂)+(pi-1/w₃)-t]

Where D(t) varies continuously and is a real number, w_i (i = 1,2,3) is a frequency and has a constant value and T_i is the period (= 2pi/w_i).

Thanks in advance for your reply

Best regards

Cristina

 

[tkhunny]

Is it continuous?
Is the derivative the same from both sides?

You're almost done.