Word Count : 388

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To discuss the continuity and differentiability of the function at \( x = \frac{2}{3} \), we analyze the left-hand limit (LHL), right-hand limit (RHL), and differentiability conditions at that point. ### Step 1: Checking Continuity at \( x = \frac{2}{3} \) A function is continuous at \( x = \frac{2}{3} \) if: \[ \lim_{x \to \frac{2}{3}^-} f(x) = \lim_{x \to \frac{2}{3}^+} f(x) = f\left(\frac{2}{3}\right) \] #### Left-Hand Limit (LHL): For \( x < \frac{2}{3} \), the function is: \[ f(x) = x - \frac{2}{3} \] Taking the limit as \( x \to \frac{2}{3}^- \): \[ \lim_{x \to ____ ____ ____ _________ _______ ______ __________ _________ _________ _________ ________ ________.
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