Let f : [-1, 2] → R be given by f(x) = 2x² + x + [x²] - [x], where [t] denotes the greatest integer

Q: Why do we check $\sqrt{2}$ and $\sqrt{3}$? They aren't integers.

Great question! The greatest integer function $[g(x)]$ is discontinuous whenever $g(x)$ is an integer. In our function, we have $[x^2]$. This term will jump whenever $x^2$ hits an integer value, not just when $x$ itself is an integer. Since $x^2$ can be $1, 2, 3$ within our interval, $x$ must be $\sqrt{1}, \sqrt{2}, \sqrt{3}$. Always look at the argument of the greatest integer function to find the 'hidden' jump points.

Q: Why is the function continuous at $x=1$ but not at $\sqrt{2}$?

Continuity is a delicate balance. At $x=1$, both $[x^2]$ and $[x]$ are about to jump. However, their jumps happen to cancel each other out in the expression $f(x) = 2x^2 + x + [x^2] - [x]$. At $x=\sqrt{2}$, only $[x^2]$ jumps, while $[x]$ remains constant (it is $1$ near $\sqrt{2}$). Because there is no corresponding jump in another term to 'fix' the discontinuity, the function breaks.

Q: Do I really need to check the endpoints $x=-1$ and $x=2$?

Yes, absolutely! In the context of continuity on a closed interval $[a, b]$, we define continuity at the left endpoint $a$ by the right-hand limit, and at the right endpoint $b$ by the left-hand limit. If the limit from the inside does not equal the function value at the endpoint, the function is considered discontinuous there. Never skip the boundaries!

Question

Let $f : [-1, 2] 	o \mathbb{R}$ be given by $f(x) = 2x^2 + x + [x^2] - [x]$, where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is :

Options:

(A) 5

(B) 6

(C) 3

(D) 4

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The function is defined as $f(x) = 2x^2 + x + [x^2] - [x]$ on the interval $[-1, 2]$. This function is a blend of a smooth quadratic polynomial and two "staircase" functions.

The polynomial part, $2x^2 + x$, is perfectly continuous. It is the greatest integer terms, $[x^2]$ and $[x]$, that cause the potential discontinuities.

A greatest integer function jumps whenever its argument hits an integer. Therefore, we must investigate points where $x \in \mathbb{Z}$ or $x^2 \in \mathbb{Z}$.

### The Anatomy of the Hunt

Within the interval $[-1, 2]$, $x$ is an integer at $\{-1, 0, 1, 2\}$.

Meanwhile, $x^2$ ranges from $0$ to $4$, so $x^2$ hits integers at $0, 1, 2, 3, 4$. Solving $x^2 = n$ for $x$ gives us the set $\{0, 1, \sqrt{2}, \sqrt{3}, 2\}$.

Combining these, our list of candidate points for discontinuity is:
$$\{-1, 0, 1, \sqrt{2}, \sqrt{3}, 2\}$$

### The Rigorous Investigation

Let us test these points one by one. At $x = -1$:
$$f(-1) = 2(1) - 1 + [1] - [-1] = 2 - 1 + 1 - (-1) = 3$$

As $x 	o -1^+$, $x^2$ is slightly less than $1$, so $[x^2] = 0$, and $[x] = -1$. The limit is:
$$\lim_{x 	o -1^+} f(x) = 2(1) - 1 + 0 - (-1) = 2$$
Since $3 
eq 2$, the function is **discontinuous** at $x = -1$.

Moving to $x = 0$, we find $f(0) = 0$. The left-hand limit as $x 	o 0^-$ involves $x^2 	o 0^+$ (so $[x^2]=0$) and $x 	o 0^-$ (so $[x]=-1$):
$$\lim_{x 	o 0^-} f(x) = 0 + 0 + 0 - (-1) = 1$$
Since $0 
eq 1$, the function is **discontinuous** at $x = 0$.

Now, we examine $x = 1$:
$$f(1) = 2(1) + 1 + [1] - [1] = 3$$
The left-hand limit as $x 	o 1^-$ gives $[x^2]=0$ and $[x]=0$, resulting in $2+1+0-0 = 3$. The right-hand limit as $x 	o 1^+$ gives $[x^2]=1$ and $[x]=1$, resulting in $2+1+1-1 = 3$.

Because the limits match, the function is **continuous** at $x = 1$.

Next, we check the irrationals. At $x = \sqrt{2}$:
$$f(\sqrt{2}) = 2(2) + \sqrt{2} + [2] - [\sqrt{2}] = 4 + \sqrt{2} + 2 - 1 = 5 + \sqrt{2}$$
The left-hand limit as $x 	o \sqrt{2}^-$ involves $x^2 	o 2^-$ (so $[x^2]=1$) and $[x]=1$:
$$\lim_{x 	o \sqrt{2}^-} f(x) = 4 + \sqrt{2} + 1 - 1 = 4 + \sqrt{2}$$
Since $5 + \sqrt{2} 
eq 4 + \sqrt{2}$, the function is **discontinuous** at $x = \sqrt{2}$. A similar logic applies to $x = \sqrt{3}$, where the function also breaks.

Finally, at $x = 2$, we find the function is **continuous**.

### The Final Tally

After our investigation, we found discontinuities at the set $\{-1, 0, \sqrt{2}, \sqrt{3}\}$.

There are exactly **4 points** of discontinuity in the given interval. This problem demonstrates that continuity is determined by the precise moments where the "staircase" jumps.

Animated Solution for Mathematics - Limits, Continuity and Differentiability: Let $f : [- 1, 2] \to R$ be given by $f (x) = 2 x^{2} + x + [x^{2}] - [x]$ , where $[t]$ denotes the greatest integer less than or equal to $t$ . The number of points, where $f$ is not continuous, is :

Select Answer:

Visualized Solution

Understanding the Function $f (x)$

Identifying Candidate Points

Finding the Exact Candidate Points

Checking Continuity at $x = - 1$

Checking Continuity at $x = 0$

Checking Continuity at $x = 1$

Checking Continuity at $x = 2$

Checking Continuity at $x = 3$

Checking Continuity at $x = 2$

Final Conclusion

The Sigma Insight: Continuity at a Point and in an Interval

Analyzing the Setup

The Anatomy of the Hunt

The Rigorous Investigation

The Final Tally

Similar Questions

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Select Answer:

Visualized Solution

Understanding the Function .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(x)

Identifying Candidate Points

Finding the Exact Candidate Points

Checking Continuity at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=−1

Checking Continuity at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=0

Checking Continuity at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=1

Checking Continuity at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=2​

Checking Continuity at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=3​

Checking Continuity at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=2

Final Conclusion

The Sigma Insight: Continuity at a Point and in an Interval

Analyzing the Setup

The Anatomy of the Hunt

The Rigorous Investigation

The Final Tally

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