Make examples that answer a question

An example becomes useful when you can explain what it tests. After a new definition, resist filling a notebook with objects that merely have the right name. Choose a question: which condition is doing the work, which familiar operation preserves the property, or where might a tempting converse fail?

For a first experiment, compare subspaces of $\mathbb R^2$. The set $U=\{(x,y):x+y=0\}$ contains zero and is closed under linear combinations. The parallel line $L=\{(x,y):x+y=1\}$ misses zero. That is already a decisive failure, so there is no need to check every subspace axiom for $L$. Moving a line a small distance has preserved its appearance while changing its algebraic status.

Change one feature and keep a record

Write the defining equation, the property being tested, and the decisive calculation. Then make a nearby object. Replace an equality by an inequality, allow an endpoint, or remove a finiteness condition. Change one feature at a time so that the comparison has an explanation.

For instance, $x\mapsto x^2$ is continuous on $\mathbb R$ but is not uniformly continuous there. Take $x_n=n$ and $y_n=n+1/n$. Their distance tends to zero, while

Uniform continuity would force these output distances to tend to zero: apply its single $\delta$ to both inputs once $1/n<\delta$. On a bounded closed interval the same function is uniformly continuous. The domain belongs in the example's name; “the square function” alone leaves out the relevant information.

A failed construction can be informative too. If you seek a continuous function on $[0,1]$ that is unbounded, the extreme value theorem rules it out. Record the obstruction. Searching longer will not produce the requested object; changing the domain or dropping continuity might.

Test the quantifiers with a moving point

To distinguish pointwise and uniform convergence, try

At $x=0$ the value is zero. At each fixed $x>0$, dividing numerator and denominator by $n^2$ shows convergence to zero. But at the moving point $x=1/n$, the value is always $1/2$. Thus the convergence is not uniform. The inequality $2t\leq1+t^2$ for $t\geq0$ also shows that the supremum is exactly $1/2$.

This example earns its space because it identifies a specific trap: checking each fixed input does not control all inputs at once. Jiří Lebl's Basic Analysis, §6.1, especially §§6.1.2–6.1.3, gives the definitions and uniform norm language needed to state that distinction precisely.

Keep a small collection organized by questions rather than prestige: a map with a large kernel, a continuous function with an inconvenient domain, a sequence whose bad behavior moves. Add enough verification that another reader could check the claim. Ravi Vakil's learning advice, in the opening discussion of learning theory, recommends calculating basic cases. The practical extension here is to keep the calculation attached to the question it resolves.