Question: a mathematical question about a continuous function and Integral and differentiability

on the basis of Fundamental theorem of calculus, the function " f " has Antiderivative F , whichi F is continuous and differentiabe in the range of integration, i have a question ? if function f has an Antiderivative F, should it be also continuous and differentiabe in that range? in the picture below it says "when f is continuous", can f be discontinuous? how it can be ? any examples?

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