Real Analysis/Limits and Continuity Exercises
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These problems are on the difficult or, to put it differently if not mildly, non-standard type. Try to work the problems without the hints because most times, you might have a different approach or way of thinking about a problem. Use the hints only if you are truly stuck! Without further ado, here are the problems:
- Prove that the function, f(x) = 1/x is not uniformly continuous on the interval (0,∞).
- Prove that a convex function is continuous (Recall that a function is a convex function if for all and all with , )
- Prove that every continuous function f which maps [0,1] into itself has at least one fixed point, that is such that
- Prove that the space of continuous functions on an interval has the cardinality of
- Let be a monotone function, i.e. . Prove that has countably many points of discontinuity.
- Let be a differentiable function, and suppose there is some positive constant such that for all . (a) Prove that is Lipschitz continuous on (Hint: Use the mean value theorem). (b) Show that every function which is Lipschitz continuous is also uniformly continuous (and therefore the function you are working with is uniformly continuous).