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The assertion states that if a function is continuous on a closed interval, it must be differentiable on that interval. We must test this against known counterexamples in calculus.
Consider the function f(x) = |x| on the interval [-1, 1]. This function is continuous everywhere on the interval, including at x = 0. However, the derivative at x = 0 does not exist because the left-hand derivative is -1 and the right-hand derivative is 1. Thus, continuity does not imply differentiability.
The reason claims that continuity ensures smoothness, which guarantees differentiability. This is mathematically incorrect. Continuity is a necessary condition for differentiability, but it is not a sufficient condition. Smoothness is a property associated with the existence of derivatives, not a cause of them.
Since the assertion is false and the reasoning provided is also false, the statement is incorrect.
Final Answer: Both Assertion and Reason are False.