Let f: R2R where f(x, y) = x² + y² if both x and y are rational and f(x, y) = 0 otherwise. Prove that Df(0, 0) exists. Note: Df(c) only exists at c = (0,0) and f is continuous only at (0,0).
Let f: R2R where f(x, y) = x² + y² if both x and y are rational and f(x, y) = 0 otherwise. Prove that Df(0, 0) exists. Note: Df(c) only exists at c = (0,0) and f is continuous only at (0,0).
Let f: R2R where f(x, y) = x² + y² if both x and y are rational and f(x, y) = 0 otherwise. Prove that Df(0, 0) exists. Note: Df(c) only exists at c = (0,0) and f is continuous only at (0,0).
Transcribed Image Text:Let \( f: \mathbb{R}^2 \to \mathbb{R} \) where \( f(x, y) = x^2 + y^2 \) if both \( x \) and \( y \) are rational and \( f(x, y) = 0 \) otherwise. Prove that \( Df(0,0) \) exists. Note: \( Df(c) \) only exists at \( c = (0,0) \) and \( f \) is continuous only at \( (0,0) \).
Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.
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