From: PointedEars@web.de   
      
   Paul B. Andersen wrote:   
   > But first we have to address your belief   
   > that the derivative of a function which is zero   
   > must be zero.   
   >   
   > | Den 30.01.2026 15:59, skrev Maciej Woźniak:   
   > |> Velocity "in own   
   > |> rest frame" is always 0, no acceleration.   
   > |> Sorry, trash.   
   >   
   > You have to learn some elementary calculus first.   
   >   
   > Let the speed be a function of time v(t).   
   > At the instant, t = t₀, the speed is zero. v(t₀) = 0.   
   > Let Δt be a time interval.   
   >   
   > The definition of the derivative of v(t₀) is:   
   >   dv(t₀)/dt = lim (v(t₀+Δt)-v(t₀))/Δt when Δt->0   
   >   
   > To see the result of this definition, we have to   
   > know the function.   
   > Let the function be: v(t) = k⋅(t-t₀) where k is a constant.   
   > v(t₀+Δt)-v(t₀) = k⋅(t₀+Δt-t₀)-k⋅(t₀-t₀) = k⋅Δt   
   >   
   > So the definition above becomes:   
   >   dv(t₀)/dt = lim k⋅Δt/Δt when Δt->0 = k   
   >   
   > The acceleration dv/dt when v = 0 is k.   
   >   
   > That the derivative of a function must be zero   
   > when the function is zero is indeed a weird idea.   
      
   The problem is language.  Mathematics as a language is precise, but when   
   mathematics is described using natural language, that is not necessarily so.   
      
   One must distinguish between a function that is _identically_ zero, i.e.   
   whose value is zero _everywhere_, and a function whose value is zero _for a   
   finite number of arguments in its domain_.   
      
   The derivative of the former function *is* actually zero because it is a   
   special case of a constant function, but the derivative of the latter   
   function is not necessarily zero.   
      
   > You know that sin(0) = 0   
      
   Yes.   
      
   > and dsin(0)/dt = cos(0) = 1   
      
   No.   
      
     dsin(0)/dt = d(0)/dt = 0.   
      
   But   
      
     dsin(t)/dt|_{t=0} := [dsin(t)/dt](0) = cos(0) = 1.   
      
   --   
   PointedEars   
      
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