Real Analysis/Limits and Continuity Exercises

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

Unsorted 1

 * 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
 * 2) * $$\lim_{x \rightarrow c}{(f+g)(x)} = f(c) + g(c)$$
 * 3) * $$\lim_{x \rightarrow c}{(f\cdot g)(x)} = f(c)\cdot g(c)$$
 * 4) * $$\lim_{x \rightarrow c}{\left( \dfrac{f}{h} \right)(x)} = \dfrac{f(c)}{h(c)}$$, given that $$h$$ is a function such that $$h(c) \ne 0$$
 * 5) 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 $$I$$

Unsorted 2
''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,&infin;).
 * 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 $$x \in (a,b)$$.
 * 7) Prove that $$f$$ is Lipschitz continuous on $$(a,b)$$
 * 8) Show that every function which is Lipschitz continuous is also uniformly continuous (and therefore the function $$f$$ you are working with is uniformly continuous).