Solution
### Related Formula
The greatest integer function [u]$[u]$ changes value and experiences a step discontinuity at any point where its inner argument u$u$ takes on an integer value.
### Core Logic
Analyze the potential points where either component function argument changes into an integer within the interval x in [0, 4]$x \in [0, 4]$.
1. For left[fracx^22right]$\left[\frac{x^2}{2}\right]$:
fracx^22$\frac{x^2}{2}$ can range from frac02 = 0$\frac{0}{2} = 0$ up to frac162 = 8$\frac{16}{2} = 8$.
Integer values are reached at fracx^22 = 0, 1, 2, 3, 4, 5, 6, 7, 8$\frac{x^2}{2} = 0, 1, 2, 3, 4, 5, 6, 7, 8$, which means critical test locations are:
x = 0, \, sqrt2, \, 2, \, sqrt6, \, sqrt8, \, sqrt10, \, sqrt12, \, sqrt14, \, 4$$x = 0, \, \sqrt{2}, \, 2, \, \sqrt{6}, \, \sqrt{8}, \, \sqrt{10}, \, \sqrt{12}, \, \sqrt{14}, \, 4$$
2. For [sqrtx]$[\sqrt{x}]$:
sqrtx$\sqrt{x}$ can range from sqrt0 = 0$\sqrt{0} = 0$ to sqrt4 = 2$\sqrt{4} = 2$.
Integer values are reached at sqrtx = 0, 1, 2$\sqrt{x} = 0, 1, 2$, which means critical test locations are:
x = 0, \, 1, \, 4$$x = 0, \, 1, \, 4$$
### Step 1: Audit Each Critical Point
Combine the set of test points within domain boundaries (0, 4)$(0, 4)$:
x in \1, \, sqrt2, \, 2, \, sqrt6, \, sqrt8, \, sqrt10, \, sqrt12, \, sqrt14\$$x \in \{1, \, \sqrt{2}, \, 2, \, \sqrt{6}, \, \sqrt{8}, \, \sqrt{10}, \, \sqrt{12}, \, \sqrt{14}\}$$
Let's evaluate the left and right hand limits at these specific values:
- At x = 1$x = 1$: [sqrtx]$[\sqrt{x}]$ steps up while left[fracx^22right]$\left[\frac{x^2}{2}\right]$ is constant implies$\implies$ Discontinuous.
- At x = sqrt2$x = \sqrt{2}$: left[fracx^22right]$\left[\frac{x^2}{2}\right]$ steps up while [sqrtx]$[\sqrt{x}]$ is constant implies$\implies$ Discontinuous.
- At x = 2$x = 2$: Both functions experience an simultaneous integer step. Let's inspect:
- f(2) = [2] - [sqrt2] = 2 - 1 = 1$f(2) = [2] - [\sqrt{2}] = 2 - 1 = 1$
- f(2^-) = [1.99] - [1.41] = 1 - 1 = 0$f(2^-) = [1.99] - [1.41] = 1 - 1 = 0$
Since LHL neq$\neq$ value at point, it is Discontinuous.
Continuing this verification down the full combined list confirms that none of the step jumps cancel each other out.
### Step 2: Sum the Discontinuity Points
Counting all isolated inner points within (0, 4)$(0, 4)$ yields exactly 8$8$ locations:
textTotal Points = 8$$\text{Total Points} = 8$$
### Pattern Recognition
When two greatest integer functions drop steps simultaneously at the same point (like at x=2$x=2$), always write out the explicit left and right limits manually, as simultaneous steps occasionally step in matching directions and maintain unexpected continuity.
### Evaluation Rubric / Model Answer
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### Chapter Mix
Class 12 Mathematics: Limits, Continuity and Differentiability
