Real Analysis/Limits and Continuity Exercises

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Real Analysis

These are a list of problems for the Limits and Continuity section of the wikibook.

Unsorted 1Edit

  1. Although the wikibook asserts the truth of the following questions in this table, it is a good exercise to prove them. Thus, given the continuous functions f and g, prove the following
    • \lim_{x \rightarrow c}{(f+g)(x)} =  f(c) + g(c)
    • \lim_{x \rightarrow c}{(f\cdot g)(x)} =  f(c) + g(c)
    • \lim_{x \rightarrow c}{\left( \dfrac{f}{h} \right)(x)} = \dfrac{f(c)}{h(c)}, given that h is a function such that h(c) \ne 0
  2. Given a continuous function f and g over any interval I, prove that f \circ \lim_{x\rightarrow a} g(x) = \lim_{x\rightarrow a}f \circ g(x) for all x in the interval I
Comments and Further Reading

Question 2 is the proof that a limit can "transfer" between a composition of functions.

Unsorted 2Edit

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:

  1. Prove that the function, f(x) = 1/x is not uniformly continuous on the interval (0,∞).
  2. Prove that a convex function is continuous (Recall that a function f: (a,b) \rightarrow \mathbb{R} is a convex function if for all x,y \in (a,b) and all s,t \in [0,1] with s+t = 1, f(sx+ty) \leq sf(x)+tf(y))
  3. Prove that every continuous function f which maps [0,1] into itself has at least one fixed point, that is \exists p \in [0,1] such that f(p) = p
  4. Prove that the space of continuous functions on an interval has the cardinality of \mathbb{R}
  5. Let f:[a,b] \rightarrow \mathbb{R} be a monotone function, i.e. \forall x,y \in [a,b]; x \leq y \Rightarrow f(x) \leq f(y). Prove that f has countably many points of discontinuity.
  6. Let f:(a,b) \rightarrow \mathbb{R} be a differentiable function, and suppose there is some positive constant K such that |f'(x)|\le K for all x \in (a,b).
    1. Prove that f is Lipschitz continuous on (a,b)
    2. Show that every function which is Lipschitz continuous is also uniformly continuous (and therefore the function f you are working with is uniformly continuous).
Hint (Question 6.1)

The Mean Value Theorem can be used here.