Counterexample : A counterexample to a statement of the form 'all objects of type x have property y' is an object of type x that does not have property y. A counterexample to a statement shows that the statement is not true in general. For example the number 3 is a counterexample to the statement that all numbers are even.

Function : A function is a mapping in which every element in one set is mapped to exactly one element of a second set. Most often the mapping is described using a rule. For example the function $f(x)=x+1$ maps 2 to 3 and -1 to 0.

Integral : The integral of a function $f$ between $a$ and $b$ is defined as $\int\limits_{a}^{b}f(x)dx$ $=\lim \limits_{n\to\infty} \frac{(b-a)}{n}\sum\limits_{k=1}^{n}f(a+\frac{k(b-a)}{n}).$ For example $\int\limits_{a}^{b}xdx=\frac{b^2}{2}-\frac{a^2}{2}.$ The fundamental theorem of calculus provides a simple way to calculate integrals using antiderivatives.

Sum : A sum is the result of an addition.

Counterexample : A counterexample to a statement of the form 'all objects of type x have property y' is an object of type x that does not have property y. A counterexample to a statement shows that the statement is not true in general. For example the number 3 is a counterexample to the statement that all numbers are even.

Function : A function is a mapping in which every element in one set is mapped to exactly one element of a second set. Most often the mapping is described using a rule. For example the function $f(x)=x+1$ maps 2 to 3 and -1 to 0.

Integral : The integral of a function $f$ between $a$ and $b$ is defined as $\int\limits_{a}^{b}f(x)dx$ $=\lim \limits_{n\to\infty} \frac{(b-a)}{n}\sum\limits_{k=1}^{n}f(a+\frac{k(b-a)}{n}).$ For example $\int\limits_{a}^{b}xdx=\frac{b^2}{2}-\frac{a^2}{2}.$ The fundamental theorem of calculus provides a simple way to calculate integrals using antiderivatives.

Integration : Calculating an integral is called integration.

Interval : An interval is a part of the number line between two numbers $a$ and $b.$ $a$ and $b$ can but do not have to be part of the interval. If $a$ and $b$ are part of the interval the interval is a closed interval. If $a$ and $b$ are not part fo the interval the interval is an open interval. If either $a$ or $b$ but not both are part of the interval the interval is called half-open or half-closed. $a$ can be equal to $-\infty$ and $b$ can be equal to $\infty$.

Power : A power is a number of the form $a^b.$ $b$ is called the exponent of the power and $a^b$ is called a power of $a$. For natural numbers $b$ the number $a^b$ is an abbreviation for successively multiplying $a$ by itself $b$ times. For example $2^3=2\cdot 2\cdot 2=8.$ For fractional exponents $b=\frac{p}{q}$ the number $a^{\frac{p}{q}}$ is defined as $\sqrt[q]{a^p}.$ For arbitrary real exponents $b$ the power $a^b$ is defined as the limit of $a^{b_n}$ with rational $b_n$ that converge towards $b.$

Half : A Half is the number equal to $\frac{1}{2}=0.5.$

Integral : The integral of a function $f$ between $a$ and $b$ is defined as $\int\limits_{a}^{b}f(x)dx$ $=\lim \limits_{n\to\infty} \frac{(b-a)}{n}\sum\limits_{k=1}^{n}f(a+\frac{k(b-a)}{n}).$ For example $\int\limits_{a}^{b}xdx=\frac{b^2}{2}-\frac{a^2}{2}.$ The fundamental theorem of calculus provides a simple way to calculate integrals using antiderivatives.

Limit : The limit of a function $f$ for $x$ converging to $x_0$ or $\lim\limits_{x\to 0}f(x)$ is a number $y$ such that for every $\epsilon\gt 0$ there is a $\delta\gt 0$ with $|f(x)-y|\lt\epsilon$ for all $|x-x_0|\lt\delta.$ This means that if $x$ only gets close enough to $x_0$ it will get and stay arbitrarily close to $y.$

Power : A power is a number of the form $a^b.$ $b$ is called the exponent of the power and $a^b$ is called a power of $a$. For natural numbers $b$ the number $a^b$ is an abbreviation for successively multiplying $a$ by itself $b$ times. For example $2^3=2\cdot 2\cdot 2=8.$ For fractional exponents $b=\frac{p}{q}$ the number $a^{\frac{p}{q}}$ is defined as $\sqrt[q]{a^p}.$ For arbitrary real exponents $b$ the power $a^b$ is defined as the limit of $a^{b_n}$ with rational $b_n$ that converge towards $b.$