Suppose that the function f: [0, 1] in R is continuous, f(O) > 0, and f(l) = 0.Prove that there is a number x 0 in (0, 1] such that f(x0 ) = 0 and f(x) > 0 for0 <=x < x 0 ; that is, there is a smallest point in the interval [0, 1] at which thefunction f attains the value 0.

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A: Consider the function Then and The graph of the function is shown below,

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A: This is a problem of the application of differential calculus.

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A: This is the question of Real topology, and real analysis.

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.