https://file.aiquickdraw.com/mathbot/file/1724860385320cxq5lzlv.jpg

 
(b)      Determine whether the statement is True or False. Explain your answer:                 

                        i.          If *f(x)* is continuous at *x = c*, then so is *ǀf(x)ǀ.*

                        ii.         If ǀ*f(x)ǀ* is continuous at *x = c*, the so is *f(x).*

                        iii.        If *f* and *g* are discontinuous at *x = c*, then so is *f + g*.

                        iv.       If *f(x)* is continuous at *x = c*, the so is 

The expression in the image appears to be the square root of a function \( \sqrt{f(x)} \). The question contains several statements related to continuity and the square root of a function.

Let me address the statements one by one:

### i. If \( f(x) \) is continuous at \( x = c \), then so is \( |f(x)| \).
- **True.** If \( f(x) \) is continuous at \( x = c \), then the absolute value \( |f(x)| \) is also continuous at \( x = c \). This is because the absolute value function is continuous, and the composition of continuous functions is continuous.

### ii. If \( |f(x)| \) is continuous at \( x = c \), then so is \( f(x) \).
- **False.** The continuity of \( |f(x)| \) at \( x = c \) does not necessarily imply the continuity of \( f(x) \) at \( x = c \). For example, if \( f(x) \) is a piecewise function that changes sign at \( x = c \) but remains continuous in absolute value, \( |f(x)| \) could be continuous while \( f(x) \) is not.

### iii. If \( f \) and \( g \) are discontinuous at \( x = c \), then so is \( f + g \).
- **False.** If both \( f(x) \) and \( g(x) \) are discontinuous at \( x = c \), it does not necessarily mean that their sum \( f(x) + g(x) \) will be discontinuous at \( x = c \). For example, \( f(x) \) and \( g(x) \) could have opposite discontinuities that cancel each other out, leading to a continuous sum.

### iv. If \( f(x) \) is continuous at \( x = c \), then so is \( \sqrt{f(x)} \).
- **True**, provided that \( f(x) \geq 0 \) for \( x \) near \( c \). If \( f(x) \) is continuous at \( x = c \) and \( f(c) \geq 0 \), then \( \sqrt{f(x)} \) is also continuous at \( x = c \) because the square root function is continuous for non-negative inputs.

Would you like further details on any of these points or explanations? Here are some related questions to expand your understanding:

1. What are the conditions for a function to be continuous at a point?
2. How does the continuity of \( |f(x)| \) relate to the continuity of \( f(x) \)?
3. Can you provide an example of functions \( f(x) \) and \( g(x) \) that are both discontinuous at \( x = c \), but their sum \( f(x) + g(x) \) is continuous at \( x = c \)?
4. What happens if \( f(x) \) is negative in the context of \( \sqrt{f(x)} \)?
5. Can the square root of a continuous function ever be discontinuous?

**Tip:** Always check the domain and behavior of functions when dealing with continuity, especially when applying operations like absolute value or square root.

Explore the concept of continuity in functions and the effects of operations like absolute value and square root on their continuity. This detailed explanation covers key statements about function continuity and provides insights into mathematical reasoning. Ideal for advanced mathematics enthusiasts.

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