Show that if f : [a, b] → R is continuous and ) x a f = 0 for all x ∈ [a, b], then f(x) = 0 everywhere on [a, b]. Provide an example to show that this conclusion does not follow if f is not continuous.

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Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as: