Let y = f(x) be defined in [a, b], then
(i) Test of continuity at `x = c, a lt c lt b`
(ii) Test of continuity at x = a
(iii) Test of continuity at x = b
Case I Test of continuity at `x = c, a lt c lt b`
If y = f(x) be defined at x = c and its value f(c) be equal to limit of f(x) as `x rarr c` i.e. f(c) = `lim_(x to c) f(x)`
or `lim_(x to c^(-))f(x) = f(c) = lim_(x to c^(+)) f(x)`
or LHL = f(c) = RHL
then, y = f(x) is continuous at x = c.
Case II Test of continuity at x = a
If RHL = f(a)
Then, f(x) is said to be continuous at the end point x = a
Case III Test of continuity at x = b, if LHL = f(b)
Then, f(x) is continuous at right end x = b.
Max ([x],|x|) is discontinuous at

Options

A. x = 0
B. `phi`
C. `x = n, n in I`
D. None of these

Answer

To determine the continuity of the function \( y = \max(\lfloor x \rfloor, |x|) \) on the interval \([a, b]\), we will analyze the continuity at a general point \( c \) within the interval, as well as at the endpoints \( a \) and \( b \).

### Step 1: Continuity at a General Point \( c \) (where \( a < c < b \))

To test for continuity at \( x = c \), we need to check the following condition:

\[ f(c) = \lim_{x \to c} f(x) \]

This can be broken down into left-hand limit (LHL) and right-hand limit (RHL):

\[ \lim_{x \to c^-} f(x) = f(c) = \lim_{x \to c^+} f(x) \]

If these conditions hold true, then \( f(x) \) is continuous at \( x = c \).

### Step 2: Continuity at the Left Endpoint \( a \)

To check continuity at \( x = a \), we need to ensure that the right-hand limit equals the function value at \( a \):

\[ \lim_{x \to a^+} f(x) = f(a) \]

If this condition is satisfied, then \( f(x) \) is continuous at \( x = a \).

### Step 3: Continuity at the Right Endpoint \( b \)

To check continuity at \( x = b \), we need to ensure that the left-hand limit equals the function value at \( b \):

\[ \lim_{x \to b^-} f(x) = f(b) \]

If this condition holds, then \( f(x) \) is continuous at \( x = b \).

### Step 4: Analyzing the Function \( y = \max(\lfloor x \rfloor, |x|) \)

1. **Graph of \( \lfloor x \rfloor \)**: This function is discontinuous at integer values of \( x \) because it jumps at each integer point.
2. **Graph of \( |x| \)**: This function is continuous everywhere.

### Step 5: Finding Points of Discontinuity

The function \( y = \max(\lfloor x \rfloor, |x|) \) will be discontinuous at points where \( \lfloor x \rfloor \) changes value, which occurs at each integer \( n \). Therefore, the function will be discontinuous at:

\[ x = n \quad \text{for all integers } n \]

### Conclusion

The function \( y = \max(\lfloor x \rfloor, |x|) \) is discontinuous at all integer points.