Animated Solution for Mathematics - Limits, Continuity and Differentiability: Let $a\in \mathbb{Z}$ and $[t]$ be the greatest integer $\le t$, then the number of points, where the function $f(x)=[a+13\sin x]$, $x\in (0,\pi )$ is not differentiable, is ________.

Welcome, future engineers! Today, we are going to dissect a problem that perfectly captures the elegance of the Greatest Integer Function (GIF). When you see a function like $f(x)=[a+13\sin x]$, it is easy to feel intimidated by the combination of a constant, a trigonometric function, and the GIF.

But let's take a breath and look at the soul of this expression.

First, let's address the constant $a$. We are told $a\in \mathbb{Z}$.

Remember the fundamental property of the GIF: $[x+I]=[x]+I$ for any integer $I$. Because $a$ is an integer, we can pull it right out of the bracket:

Now, think about what this means geometrically. Adding a constant $a$ simply shifts the entire graph of the function vertically.

Does shifting a graph up or down change where it has sharp corners or jumps? Absolutely not! The points of non-differentiability for $f(x)$ are identical to those of $g(x)=[13\sin x]$. We have successfully simplified our problem by stripping away the distraction.

Now, let's focus on the core: $13\sin x$ for $x\in (0,\pi )$. Imagine the graph of $y=13\sin x$.

It starts at $0$ when $x=0$, climbs smoothly to a maximum height of $13$ at $x=\frac{\pi }{2}$, and then descends back to $0$ at $x=\pi$. This is a beautiful, continuous arch.

However, we are wrapping this arch in a Greatest Integer Function. The GIF is a staircase; it stays flat at an integer value until the input hits the next integer, at which point it 'jumps' to the next level.

Therefore, the function $[13\sin x]$ will be non-differentiable exactly where $13\sin x$ hits an integer value. These are the points where the 'staircase' jumps.

We need to find all $x\in (0,\pi )$ such that $13\sin x=k$, where $k$ is an integer. Since the range of $13\sin x$ on $(0,\pi )$ is $(0,13]$, the possible integer values for $k$ are $\{1,2,3,\dots ,13\}$.

Let's draw horizontal lines at each of these integer heights. For any integer $k$ from $1$ to $12$, the line $y=k$ will intersect our sine arch twice: once while the curve is rising and once while it is falling.

Since there are $12$ such lines, we have $12\times 2=24$ points of intersection. At every one of these $24$ points, the function $[13\sin x]$ experiences a jump discontinuity, making it non-differentiable.

Finally, we must consider the special case: $k=13$. The line $y=13$ is tangent to the peak of our sine arch at $x=\frac{\pi }{2}$.

The curve does not cross this line; it only touches it. At $x=\frac{\pi }{2}$, the value of the function is $[13]=13$.

But if we move just a tiny bit to the left or right, the value of $13\sin x$ becomes slightly less than $13$ (like $12.99$), and the GIF drops to $12$. This creates an isolated point discontinuity at the peak. Because the function is discontinuous here, it is also non-differentiable.

So, we have $24$ points from the crossings of the lines $y=1$ through $y=12$, and $1$ point from the peak at $y=13$. Adding these together, we get:

The total number of points of non-differentiability is 25. It is a beautiful result, isn't it? By breaking the problem into the 'shift' and the 'staircase,' we turned a complex expression into a simple counting exercise.