VerifiedHint: A function is continuous at a point if the one-sided limits at that point exist and are equal to the function's value at that point.
Solution:
(i) The function is a first-degree polynomial. All polynomial functions are continuous over the entire real line, $\mathbb{R}$. Therefore, $f$ is continuous for all $x \in \mathbb{R}$.
(ii) For the function $f(x) = \dfrac{1}{{x-5}}$, where $x \neq 5$:
For any $k \neq 5$, we have:
$\lim_{{x \to k}} f(x) = \lim_{{x \to k}} \dfrac{1}{{x-5}} = \dfrac{1}{{k-5}}$
Also, $f(k) = \dfrac{1}{{k-5}}$ since $k \neq 5$.
Therefore, the function is continuous at every point in its domain.
(iii) For the function $f(x) = \dfrac{{x+5}}{{x^2 - 25}}$, where $x \neq -5$:
For any real $x \neq -5$, we get:
$\lim_{{x \to c}} f(x) = \lim_{{x \to c}} \dfrac{{x+5}}{{(x+5)(x-5)}} = x-5$
Thus, the function is continuous at every point in its domain.
(iv) For the function $f(x) = |x - 5|$:
This function can be expressed piecewise as:
$f(x) = \begin{cases} 5 - x & \text{if } x < 5 \\ x - 5 & \text{if } x \geq 5 \end{cases}$
For $c < 5$, $f(c) = 5 - c$ and $\lim_{{x \to c}} f(x) = 5 - c$, proving continuity for $x < 5$.
At $c = 5$, $f(c) = 0$ and both left and right limits as $x \to 5$ also equal to 0. Therefore, $f$ is continuous at $x = 5$.
For $c > 5$, $f(c) = c - 5$ and $\lim_{{x \to c}} f(x) = c - 5$, proving continuity for $x > 5$.
Thus, the function is continuous for all real numbers.
In summary, each function given has been proven to be continuous over its domain.
Additional Information:
The continuity of a function can be checked using the three conditions listed above.
A function can be continuous at some points and discontinuous at others.
There are many different types of discontinuities, including jump discontinuities, removable discontinuities, and essential discontinuities.
Note: The continuity of a function is an important property that has many applications in calculus and analysis. For example, the continuity of a function is necessary for the existence of the derivative of the function.
