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Does the close price of a candlestick necessarily have to coincide with the opening price of the next candlestick?

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No it does not. Candlesticks really have nothing to do with this, a stock price can open different then the previous day's close.

Examining the chart of TSLA provides an example it closed on 1/18/17 at 238.8 it opened on 1/19/17 at 243.7

In candlestick parlance is is known as a "gap up".

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