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.