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A function being continuous at a point means that the twosided limit at that point exists and is equal to the function's value. Point/removable discontinuity is when the twosided limit exists but isn't equal to the function's value. Jump discontinuity is when the twosided limit doesn't exist because the onesided limits aren't equal. Asymptotic/infinite discontinuity is when the twosided limit doesn't exist because t's unbounded.

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