Find the range of a univariate function over a specific interval of the variable

FunctionRange({-a*
   Log(1 - (a - a E^(-1 - 1/a + ProductLog(E^(1 + 1/a))))/a), 
  a > 0}, a, y)

5.14988*10^-9 <= y <= 0.5

and warming message:

Unable to find the exact range. Returning bounds on the range
computed using numeric optimization methods

So we have to consider Derivative

FunctionRange({D(-a*
     Log(1 - (a - a E^(-1 - 1/a + ProductLog(E^(1 + 1/a))))/a), a) // 
   Evaluate, a > 0}, a, y)

y > 0

It means that v(a) is a monotonic function ! So we just need to calculate the Limit just as @Akku14 have done.

Limit(-a*Log(1 - (a - a E^(-1 - 1/a + ProductLog(E^(1 + 1/a))))/a), 
 a -> 0, Direction -> "FromAbove")
Limit(-a*Log(1 - (a - a E^(-1 - 1/a + ProductLog(E^(1 + 1/a))))/a), 
 a -> ∞)

0

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