Real numbers, limits, and integration.
Answer five questions about Real Analysis and get instant feedback.
What means that a tiny change in input causes only a tiny change in output?
One way to picture continuity is that you can draw the graph without lifting your pencil, and it guarantees that nearby inputs always give outputs that stay close too.
What means a sequence of functions can get closer and closer to one target function?
There are different kinds of convergence, like pointwise versus uniform, and uniform convergence is strong enough that you can often swap limits with integrals or derivatives safely.
What kind of function on a closed interval $[a,b]$ must be uniformly continuous?
This is really a compactness phenomenon: on $[a,b]$ you can cover the interval by finitely many local continuity neighborhoods, unlike on $(0,1)$ where $f(x)=\frac{1}{x}$ is continuous but not uniformly continuous.
This image question appears in the interactive quiz.
A discontinuity marks where a function’s values fail to connect smoothly.
What measure on $\mathbb{R}$ is translation-invariant and gives every interval $[a,b]$ the value $b-a$?
Lebesgue measure is uniquely characterized (up to scaling) by translation-invariance and countable additivity on Borel sets, and it makes nonmeasurable sets possible via Vitali's construction.
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