How to find continuity and differentiability of a function which includes Greatest Integer Function?

Question

Lets consider function $ f(x) = [x] $ where $[\space]$ denotes a greatest Integer function. let $ g(x)$ be an arbitrary function.

How do we determine whether the function $f(g(x))$ and $g(f(x))$ is continuous and differentiable in certain interval or point.

Here's my approach, I try plotting graphs but it become tedious and difficult when function $g$ becomes complex. Does anybody know any simpler trick or algorithm to solve these type of question?

thanks.

Answer 1

Continuity and differentiability are properties of a function at a specific point rather than properties of a function as a whole.

So the "greatest integer less than or equal to $x$" function, which is usually written as $f(x) = \lfloor x \rfloor$, is continuous at all points apart from integer values of $x$. It is also differentiable, with $f'(x)=0$, at all points apart from integer values of $x$.

The compound function $f(g(x))$ is continuous at $x$ if $g$ is continuous at $x$ and $f$ is continuous at $g(x)$. It is also differentiable at at $x$ if $g$ is differentiable at $x$ and $f$ is differentiable at $g(x)$. And wherever $f$ and $g$ are differentiable the derivative of $f(g(x))$ is $f'(g(x))g'(x)$ - this is the chain rule of differentiation.

Note that these conditions are sufficient for $f(g(x)$ to be continuous/differentiable, but they are not necessary; $f(g(x))$ may be continuous/differentiable at other points as well. For example

$f(g(x)) = \lfloor x^2 \rfloor$

is continuous and differentiable at $x=0$ even though $f$ is not continuous or differentiable at $g(0)$.

Answer 2

See the box value of any function will be the same for the range $n$ to $n+1$ where $n$ is definitely an integer therefore the graph will look like steps.