Assume the function f(x, t) is continuous on the rectangle D = {(x, t) : a ≤ x ≤ b, c ≤ t ≤ d}. Explain why the function F(x) =* dcf(x, t)dt is properly defined for all x ∈ [a, b]. It should not be too surprising that Theorem 4.4.7 has an analogue in the R2 setting. The set D is compact in R2, and a continuous function on D is uniformly continuous in the sense that the δ in Definition 8.4.4 can be chosen independently of the point (x0, t0).

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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.