Let f : R → R be a function and M E R, such that for every (x, y) E R², the function g(t):= f(xt, yt) is differentiable and |g'(1)| < M. (a) Show that f is continuous at (0,0). (b) Find an example of such an f that is discontinuous at every other point of R2. Hint: Think back to how we constructed a nowhere continuous function on [0, 1].

Let f : R → R be a function and M E R, such that for every (x, y) E R², the function g(t):= f(xt, yt) is differentiable and |g'(1)| < M. (a) Show that f is continuous at (0,0). (b) Find an example of such an f that is discontinuous at every other point of R2. Hint: Think back to how we constructed a nowhere continuous function on [0, 1].

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.3: More On Functions; Piecewise-defined Functions

Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...

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