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Given a piecewise function f[t_]:=Piecewise[{{Exp[-1/t^2],t>0},{0,x<=0}}], I can define another function like this: g[t_]:=f[t+1]f[1-t]. Both the functions can be calculated and plotted. But when I define the third function h[t_]:=Integrate[g[x],{x,-:inf:,t}], h[t] cannot be used to plot. Does the definition for h[t] have anything wrong?

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    $\begingroup$ Please be more specific about what you mean by "cannot be used". Also: did you mean to write x <= 0 or t <= 0 in f? $\endgroup$ – Szabolcs Jan 29 '15 at 3:46
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  • $\begingroup$ Recommend that the definition of g be modified to g[t_] = f[t + 1] f[1 - t] // PiecewiseExpand // Simplify $\endgroup$ – Bob Hanlon Jan 29 '15 at 5:15
  • $\begingroup$ Thanks, I have revised the question. $\endgroup$ – Steven_Wang Jan 29 '15 at 7:08
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You have an error in the definition of f[] and it's faster to use NIntegrate[]:

f[t_] := Piecewise[{{Exp[-1/t^2], t > 0}, {0, t <= 0}}]
g[t_] := f[t + 1] f[1 - t]
h[t_] := NIntegrate[g[x], {x, -Infinity, t}]
Plot[h[t], {t, -1, 1}]

Mathematica graphics

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