Animated Solution for Mathematics - Limits, Continuity and Differentiability: Let $f:[-1,3]\to R$ be defined as $f(x)=[x]+[x],-1\le x<1;x+[x],1\le x<2;x+[x],2\le x\le 3$. Then, f is discontinuous at:

Imagine you are standing at the base of a grand, mathematical staircase. This is the world of the Greatest Integer Function, denoted as $[x]$.

It is a function that refuses to be smooth; it prefers to jump, to break, and to surprise us at every integer. Today, we are going to dissect a piecewise function $f(x)$ defined on the interval $[-1,3]$ and uncover exactly where it breaks.

This isn't just about finding points; it is about understanding the rhythm of the function.

Our function $f(x)$ is defined in three distinct acts:

1. For $-1\le x<1$, we have $f(x)=[x]+[x]=2[x]$. 2. For $1\le x<2$, we have $f(x)=x+[x]$. 3. For $2\le x\le 3$, we have $f(x)=x+[x]$.

Our mission is to check the continuity at the integer boundaries: $x=0,1,2,3$.

The Jump at $x=0$

Let us look at the interval around zero. For $x\in [-1,0)$, $[x]=-1$, so:

As we approach $x=0$ from the left, the function is stuck at $-2$.

But the moment we step into the interval $[0,1)$, $[x]$ becomes $0$, and $f(x)=0+0=0$. Since the left-hand limit is $-2$ and the function value $f(0)=0$, we have a discontinuity at $x=0$.

The Shift at $x=1$

Now, we move to $x=1$. From the left, in the interval $[0,1)$, we saw $f(x)=0$.

As soon as we hit $x=1$, the definition changes to $f(x)=x+[x]$. At $x=1$, $[x]=1$, so:

The left-hand limit was $0$, but the right-hand limit (and the function value) is $2$. The graph leaps from $0$ to $2$, confirming a discontinuity at $x=1$.

The Climb at $x=2$

As we approach $x=2$ from the left, we are in the interval $[1,2)$, where $f(x)=x+1$. The limit is:

However, for $x\in [2,3]$, the definition is $f(x)=x+[x]$. At $x=2$, the function value is:

The limit from the left is $3$, but the value at $x=2$ is $4$. The graph breaks again, marking a discontinuity at $x=2$.

Finally, we reach the edge of our domain at $x=3$. The function approaches $3$ from the left using the rule $f(x)=x+2$, giving a limit:

But at exactly $x=3$, the rule is:

The limit is $5$, but the value is $6$. Even at the endpoint, the function fails to be continuous, resulting in a discontinuity at $x=3$.

We have methodically checked $x=0,1,2,3$ and found a break at every single one. There are 4 points of discontinuity in total.

This problem beautifully illustrates how piecewise definitions and the Greatest Integer Function can create a landscape of jumps. By breaking the problem into these manageable intervals, we transformed a daunting task into a clear, logical journey.