Composite function : The composite function $h=f\circ g$ is defined by $h(x)=f(g(x)).$ For example for the functions $f(x)=x^2$ and $g(x)=x+1$ the composite function $h=f\circ g$ is the function $h(x)=(x+1)^2.$ The composite function of two continuous functions is continuous. The composite function of two differentiable functions is differentiable and the derivative can be calculated using the chain rule.

Continuous : A function is continuous at a point $x_0$ if $\lim \limits_{x\to x_0} f(x)=f(x_0).$

Continuous function : A function is continuous if it is continuous at every point $x.$ The graph of a continuous function does not have any jumps.

Product : A product is the result of a multiplication.

Sum : A sum is the result of an addition.

Continuous : A function is continuous at a point $x_0$ if $\lim \limits_{x\to x_0} f(x)=f(x_0).$

Continuous function : A function is continuous if it is continuous at every point $x.$ The graph of a continuous function does not have any jumps.

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 $y.$

Continuous : A function is continuous at a point $x_0$ if $\lim \limits_{x\to x_0} f(x)=f(x_0).$

Continuous function : A function is continuous if it is continuous at every point $x.$ The graph of a continuous function does not have any jumps.

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 $y.$