Count Discontinuity Points in GIF Function

❓Question

The number of points of discontinuity of the function

$$f(x)=\left[\frac{x^2}{2}\right]-[\sqrt{x}],x\in [0,4],$$

where $[\cdot ]$ denotes the greatest integer function, is equal to ?

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🖼️ Question Image

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✍️ Short Solution

For functions involving GIF (floor), remember this golden rule:

👉 Discontinuity occurs when the expression inside [ ] crosses an integer.

We’ll find all such points for each term, then combine carefully.

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🔹 Step 1 — Discontinuity points of $\left[\dfrac{x^2}{2}\right]$

This term jumps when:

$$\frac{x^2}{2} = k, \quad k\in\mathbb{Z}$$

Given $x\in[0,4]$:

$$\frac{x^2}{2}\in[0,8]$$

So possible integers:

$$k=0,1,2,3,4,5,6,7,8$$

Corresponding $x$-values:

$$x=\sqrt{2\mathrm{k}}$$

Within $[0,4]$, these are:

$$\sqrt{0},\sqrt{2},2,\sqrt{6},2\sqrt{2},\sqrt{10},\sqrt{12},\sqrt{14},4$$

So 9 potential jump points from this term.

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🔹 Step 2 — Discontinuity points of $[\sqrt{x}]$

This term jumps when:

$$\sqrt{x}=n,n\in \mathbb{Z}$$

Given $x\in[0,4]$:

$$\sqrt{x}\in [0,2]\Rightarrow n=0,1,2$$

Corresponding $x$-values:

$$x=0,\ 1,\ 4$$

So 3 potential jump points from this term.

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🔹 Step 3 — Combine the points (IMPORTANT 🔥)**

The function is:

$$f(x)=\left[\frac{x^2}{2}\right]-[\sqrt{x}]$$

Key observation:

• A jump in either term generally causes a discontinuity in $f(x)$
  

• Unless both jump by the same amount at the same point (rare but must be checked)

Let’s list all candidate points (union of both sets):

From Step 1:

$$0,\sqrt{2},2,\sqrt{6},2\sqrt{2},\sqrt{10},\sqrt{12},\sqrt{14},4$$

From Step 2:

$$0,\ 1,\ 4$$

Union gives:

$$0,1,\sqrt{2},2,\sqrt{6},2\sqrt{2},\sqrt{10},\sqrt{12},\sqrt{14},4$$

Total = 10 distinct points.

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🔹 Step 4 — Check overlapping jumps

At $x=0$:

• $\left[\frac{x^2}{2}\right]$ jumps

• $[\sqrt{x}]$ also jumps
  👉 Net jump ≠ 0 ⇒ discontinuity

At $x=4$:

• $\left[\frac{x^2}{2}\right]$ jumps

• $[\sqrt{x}]$ also jumps
  👉 Net jump ≠ 0 ⇒ discontinuity

At other points, only one term jumps ⇒ discontinuity.

So none cancel out.

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✅ Final Answer

$$\boxed{{\displaystyle \mathrm{10}}}$$

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