If not stated otherwise, we assume that the metric spaces \mathbb{R} and \mathbb{Q} are quipped with the Euclidean metric d(x,y)=|x-y|.
Exercise: Prove using the definition of continuity that the function f:[0,\infty)\rightarrow\mathbb{R} defined by f(x)=\sqrt{x} is continuous at every point p\in[0,\infty).
Proof. Suppose \epsilon>0. Let \delta=\epsilon^2. If |p-q|<\delta, then |f(p)-f(q)|^2\leq|\sqrt{p}-\sqrt{q}||\sqrt{p}+\sqrt{q}|=|p-q|<\epsilon^2, i.e., |f(p)-f(q)|<\epsilon.
Exercise: Consider the function f:\mathbb{R}\rightarrow\mathbb{R} defined by
f(x)=\bigg\{{\begin{array}{cl}1/2^k&\text{if }|x|\in[1/2^k,1/2^{k-1}),\text{ for some } k\in\mathbb{Z},\\0&\text{if }x=0\end{array}}
Prove that f is continuous at x=0.
Proof. Let g(x)=0 and h(x)=|x|. Since g(x)\leq f(x)\leq h(x) and g(0)=f(0)=h(0), by problem 5 and the continuity of g(x) and h(x), we know that f is continuous at 0.
Exercise: Suppose f:\mathbb{R}\rightarrow\mathbb{R} satisfies \lim_{h\rightarrow 0}[f(x+h)-f(x-h)]=0 for every x\in\mathbb{R}. Does this imply f is continuous? Give a proof or a counterexample.
Proof. Suppose f(x) = 0 for all x\in\mathbb{R} but 0, and f(0)=1. Then f satisfies the requirement, but fails to be continuous.
Exercise: Suppose (X,d) is a metric space and f,g:X\rightarrow\mathbb{R}. State whether the following statements are true or false. If the statement is true, prove it. If it is false, give a counterexample.
- If f is continuous on X, then |f| is continuous on X, where |f|(x)=|f(x)| for all x\in X.
- If |f| is continuous at p\in X, then f is continuous at p.
- Define h: X\rightarrow\mathbb{R} by h(x)=\max\{f(x),g(x)\} for all x\in X. If f and g are continuous at p\in X, then h is continuous at p.
- If f is continuous at p\in X, and if f(p)>M for some M\in\mathbb{R}, then there is a neighborhood N of p in X, such that f(x)>M for all x\in N.
- Suppose d is the discrete metric on X. Then every map f:X\rightarrow\mathbb{R} is continuous on X.
Proof.
- Since |f| is really the composition of two continuous functions, i.e., the function f followed by the absolute value function, so it’s continuous.
- Suppose X=\mathbb{R} and f(x)=1 if x\geq 0, otherwise f(x)=-1. Then |f| is a constant function, which is continuous, but f itself fails to be continuous at x=0.
- As h(x)=\max\{f(x),g(x)\}=(f(x)+g(x)-|f(x)-g(x)|)/2, h is continuos.
- Consider the pre-image of (M,\infty]. Since its an open set containing f(p), by the continuity of f at p, the pre-image is a neighborhood of p. Let N=f^{-1}(M,\infty]. It works.
- Choose your favorite map f:X\rightarrow\mathbb{R}. Whatever x\in X and \epsilon>0 are, always pick \delta=1. If |x-x'|<\delta=1, then x=x', hence f(x)-f(x')=0<\epsilon.
Exercise: Suppose f,g and h are functions from \mathbb{R} to \mathbb{R} satisfying g(x)\leq h(x)\leq f(x) for all x\in\mathbb{R}. Further, suppose f and g are continuous at x=a and f(a)=g(a). Then, prove that h is continuous at x=a.
Proof. Let b=f(a)=g(a)=h(a). As f and g are continuous at x=a, for every \epsilon > 0, there are \delta_1 > 0 and \delta_2 > 0, such that if |x-a|<\delta_1, then |f(x)-b|<\epsilon, and if |x-a|<\delta_2, then |g(x)-b|<\epsilon. Let \delta be the minimum of \delta_1 and \delta_2. If |x-a|<\delta, then h(x)-b\leq f(x)-b<\epsilon and b-h(x)\leq b-g(x)<\epsilon. This finishes the proof of the continuity at a.