Why is there still a small amount of beta-galactosidase present in E. Coli, even when no lactose is present?

Answer:
This is the basal level of expression. The repressor is not continuously bound the the operator- it dissociates every now and then, giving Polymerase a chance to transcribe the gene. It happens with every gene.

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It is hereditary, so they have it there if they need to use it later when they encounter lactose.

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It is a leaky system. The repressor does not completley inhibit transcription.

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