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Is is possible to use NDSolve with DiscreteVariables if there are no continuous-time variables?

This fails:

NDSolve[{x[0] == 1,
    WhenEvent[Mod[t, 1] == 0, x[t] -> -x[t]]}, {x}, {t, 0, 7}, 
    DiscreteVariables -> {x}]

It works if I remove the DiscreteVariables, replacing it with $x'(t)=0$:

NDSolveValue[{x[0] == 1, x'[t] == 0,
    WhenEvent[Mod[t, 1] == 0, x[t] -> -x[t]]}, {x}, {t, 0, 7}]

(This seems to defeat the purpose of DiscreteVariables.)

It works if I introduce a "dummy" continuous-time variable:

NDSolveValue[{y'[t] == 0, y[0] == 0, x[0] == 1, 
    WhenEvent[Mod[t, 1] == 0, x[t] -> -x[t]]}, {x, y}, {t, 0, 7}, 
    DiscreteVariables -> {x}]

(This seems like a hack.)

I ask because I have a system of equations

$$x_i(T_i^{k+1})=\sum_{j=1}^N a_{i,j} x_j(T_i^k), \quad i=1,\ldots,N$$

where each variable $x_i(t)$ changes only at the discrete times $\{T_i^k\}_{k \in \mathbb{N}}$.

Thoughts?

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    $\begingroup$ I often use the dummy-variable "hack" to drive DAEs. -- Might be considered a design flaw, or one might think the D in NDSolve should mean something. :) $\endgroup$ – Michael E2 May 18 '16 at 0:40
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    $\begingroup$ I see what you mean, but if NDSolve isn't the right tool for this, is there a more appropriate function I'm overlooking? $\endgroup$ – ConvexMartian May 18 '16 at 3:52
  • $\begingroup$ I think it's the best available tool, even if it's not the best imaginable tool. NDSolve is the only built-in way to construct a discontinuous InterpolatingFunction (see here for a manual method). If if one knows ahead of time the times that the variable changes, one could directly construct a Piecewise function (see here or here).... $\endgroup$ – Michael E2 May 18 '16 at 12:07
  • $\begingroup$ ...I view your problem as an "differential-order-0" form of "integration" (in the sense of putting together a function). FunctionInterpolation takes a global approach like NIntegrate. NDSolve takes a local approach, but primarily for order > 0; however, via DAEs, it can simultaneously do order-0 integrations. I don't think there's an analog of FunctionInterpolation. $\endgroup$ – Michael E2 May 18 '16 at 12:07
  • $\begingroup$ I see. Thank you @MichaelE2 ! $\endgroup$ – ConvexMartian May 18 '16 at 13:35

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