Hi Valerie, if your starter has been rising consistently and you've been cultivating it for a month, I doubt what you're seeing is mold and I think it's certainly worth a try to salvage it. I would remove some of the starter from the bottom of the container and give that a few days of twice a day feedings at room temperature. If the black spots don't return and your starter is doubling in size consistently with a pleasant aroma, then your starter is back on track and ready for baking again! While it's generally not necessary to wash out your container with each feeding, I would give your container with the spots on the side a good cleaning in very hot water, and allow it to air dry before using it again. 

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