Animated Solution for Mathematics - Definite Integration: Let $f:[0,\infty )\to [0,\infty )$ be defined as $f(x)={\int }_0^x[y]dy$ where $[x]$ is the greatest integer less than or equal to $x$. Which of the following is true?

Imagine standing at the origin of a coordinate plane, ready to embark on a journey of accumulation. We are given a function $f(x)={\int }_0^x[y]dy$, where $[y]$ is the Greatest Integer Function.

The integrand, $g(y)=[y]$, is a classic step function. It stays flat, then jumps, stays flat, then jumps again. Your task is to find the area under this staircase as you walk along the $x$-axis.

Let's address the first major hurdle: continuity. A common trap is to assume that because the integrand $[y]$ is discontinuous at every integer, the integral $f(x)$ must also be discontinuous.

But think about the physical reality of an integral. An integral is the accumulated area. As you move from $x=0.9$ to $x=1.1$, the height of the function jumps from $0$ to $1$.

However, this jump happens over an infinitesimal width. The area added during that jump is zero. Therefore, the total accumulated area $f(x)$ does not jump; it connects seamlessly at every integer. Thus, $f(x)$ is continuous everywhere on $[0,\infty )$.

Now, we turn to the question of differentiability. This is where the Fundamental Theorem of Calculus (FTC) comes into play. The FTC tells us that if $f(x)={\int }_0^{x}g(t)dt$, then $f^{\prime }(x)=g(x)$, provided $g$ is continuous.

For all non-integer points, the step function is perfectly flat and continuous. Therefore, at any non-integer $x$, the derivative $f^{\prime }(x)$ is simply $[x]$. The function is smooth and differentiable here.

But what happens at the integers? This is the heart of the problem. At an integer $x=n$, the function $[y]$ jumps. This jump in the integrand translates to a jump in the slope of our area function $f(x)$.

To prove this, we look at the definition of the derivative:

If we approach from the left ($h\to 0^{-}$), the slope is determined by the value of the function just before $n$, which is $n-1$. If we approach from the right ($h\to 0^{+}$), the slope is determined by the value just after $n$, which is $n$.

Since the left-hand derivative ($n-1$) does not equal the right-hand derivative ($n$), the function has a sharp corner—a 'kink'—at every integer. As we know, a function with a kink is not differentiable at that point.

We have walked through the geometry of the staircase. We have seen that while the area accumulates continuously, the rate of that accumulation—the slope—changes abruptly at every integer.

We have successfully navigated the trap of assuming that discontinuity in the integrand implies discontinuity in the integral. We have also rigorously confirmed that the lack of continuity in the integrand at integers destroys differentiability at those specific points.

You have mastered the behavior of the Greatest Integer Function under the lens of calculus. Keep this geometric intuition close; it is the key to solving even more complex problems in your JEE journey.