Animated Solution for Mathematics - Limits, Continuity and Differentiability: Let $[.]$ denote the greatest integer function and $f(x)=[{\tan }^{2}x]$, then:

Imagine you are standing on a vast, undulating landscape defined by the function $g(x)={\tan }^{2}x$. As you look toward the origin, $x=0$, you see the curve gently touching the x-axis, perfectly symmetric, rising gracefully on either side.

Now, we introduce the Greatest Integer Function, $[\cdot ]$, often called the floor function. It is a mathematical gatekeeper, rounding every value down to the nearest integer.

Many students fear this function because it creates sharp, sudden jumps—discontinuities that seem to break the smoothness of calculus. But today, we are going to see that even in the presence of such a function, there is a pocket of perfect, predictable calm.

To understand $f(x)=[{\tan }^{2}x]$ at $x=0$, we must first understand the behavior of ${\tan }^{2}x$ near the origin. We know that $\tan (0)=0$, so ${\tan }^2(0)=0$.

As we move slightly away from zero, ${\tan }^{2}x$ begins to increase. The crucial question is: how far can we go before the Greatest Integer Function forces a jump?

The function $[t]$ jumps whenever $t$ hits an integer. The smallest positive integer is $1$. So, we ask: when does ${\tan }^{2}x=1$?

Solving this, we find $\tan x=\pm 1$, which occurs at $x=\pm \frac{\pi }{4}$. This defines our territory: the open interval $(-\frac{\pi }{4},\frac{\pi }{4})$.

Within this interval $(-\frac{\pi }{4},\frac{\pi }{4})$, the value of ${\tan }^{2}x$ is trapped. It starts at $0$ and grows, but it never reaches $1$.

Mathematically, for all $x$ in this neighborhood, we have:

Now, apply the Greatest Integer Function. By definition, for any $t$ such that $0\le t<1$, the greatest integer $[t]$ is exactly $0$.

This is the moment of revelation! In this entire neighborhood, our function $f(x)=[{\tan }^{2}x]$ is not a complex, jumping curve; it is simply the constant function $f(x)=0$.

Now that we have simplified the function, the calculus becomes trivial and beautiful. To find the limit ${\lim }_{x\to 0}f(x)$, we look at the neighborhood of $0$.

Since $f(x)=0$ for all $x$ in $(-\frac{\pi }{4},\frac{\pi }{4})$, the limit is clearly $0$.

For continuity, we check if ${\lim }_{x\to 0}f(x)=f(0)$. We know:

Since the limit is $0$ and the function value is $0$, the function is continuous at $x=0$.

Finally, for differentiability, we look at the slope. Since the function is a constant $0$ in this neighborhood, its graph is a flat horizontal line.

The derivative of a constant is $0$. Thus, $f^{\prime }(0)=0$.

We have navigated the potential traps of the Greatest Integer Function and found that at the origin, everything is smooth, continuous, and perfectly defined. Mathematics often hides its simplicity behind complex notation, but with a little visualization, the truth reveals itself.