Animated Solution for Mathematics - Limits, Continuity and Differentiability: If $f(x)=[x]-[x/4],x\in R$, where $[x]$ denotes the greatest integer function, then :

Welcome, future engineer. Today, we are going to dissect a problem that seems deceptively simple but hides a profound lesson about the nature of continuity. We are looking at the function $f(x)=[x]-[\frac{x}{4}]$.

At first glance, it looks like a standard algebraic expression, but the presence of the Greatest Integer Function (GIF) transforms it into a staircase of logic. Let us begin our journey by understanding what continuity truly means.

Imagine a bridge. For a bridge to be continuous at a point $x=4$, you must be able to walk from the left side, cross the point $x=4$, and continue to the right side without ever lifting your foot. Mathematically, this means the left-hand limit, the right-hand limit, and the value of the function at that point must all converge to the exact same destination. If they do not, the bridge is broken, and the function is discontinuous.

Approaching from Above

Let us start by approaching $x=4$ from the right. We define this as $x=4+h$, where $h$ is a tiny, positive value approaching zero. We need to evaluate:

First, look at $[x]$. Since $x$ is slightly larger than $4$ (like $4.001$), the greatest integer less than or equal to $x$ is clearly $4$.

Now, look at the second term: $[\frac{x}{4}]$. Substituting $x=4+h$, we get:

Since $\frac{h}{4}$ is a tiny positive number, $1+\frac{h}{4}$ is slightly larger than $1$. The greatest integer of any number in the interval $[1,2)$ is $1$. So, our right-hand limit becomes $4-1=3$. We have successfully crossed the bridge from the right!

The Tricky Descent

Now, let us approach from the left. This is where most students stumble, so take a deep breath. We set $x=4-h$. We need to evaluate:

First, look at $[x]$. Since $x$ is slightly less than $4$ (like $3.999$), the greatest integer less than or equal to $x$ is $3$. This is the first trap; do not let your intuition default to $4$.

Next, look at $[\frac{x}{4}]$. Substituting $x=4-h$, we get:

Here is the second trap: $1-\frac{h}{4}$ is slightly less than $1$ (like $0.999$). The greatest integer of any number in the interval $[0,1)$ is $0$. Therefore, the second term becomes $0$. Our left-hand limit is $3-0=3$.

We have the right-hand limit as $3$ and the left-hand limit as $3$. Now, we check the functional value at $x=4$:

Since the left-hand limit, the right-hand limit, and the functional value are all equal to $3$, the bridge is perfectly intact. The function is continuous at $x=4$.

This problem teaches us that even when functions involve the 'jumpy' nature of the Greatest Integer Function, they can still harmonize to create a continuous path. Never fear the GIF; respect its steps, calculate carefully, and you will always find the truth.