1. The problem asks us to understand the definition and properties of continuity of a function at a point $a$. 2. A function $f$ is continuous at $a$ if three conditions are met: - $f(a)$ is defined (meaning $a$ is in the domain of $f$). - The limit $\lim_{x \to a} f(x)$ exists. - The limit equals the function value: $\lim_{x \to a} f(x) = f(a)$. 3. If any of these conditions fail, $f$ is discontinuous at $a$. 4. The problem states two simple functions: - $f(x) = c$, a constant function. - $f(x) = x$, the identity function. 5. For $f(x) = c$, the limit as $x$ approaches any $a$ is $c$ because the function value never changes. So, $$\lim_{x \to a} c = c$$ and since $f(a) = c$, the function is continuous everywhere. 6. For $f(x) = x$, the limit as $x$ approaches $a$ is simply $a$: $$\lim_{x \to a} x = a$$ and since $f(a) = a$, this function is also continuous everywhere. 7. These examples illustrate the general idea that simple functions like constants and identity functions are continuous at every point because their limits match their function values at all points. 8. In summary, continuity means no sudden jumps or breaks in the function at the point $a$, and these two functions are perfect examples of continuous functions. Final answer: Both $f(x) = c$ and $f(x) = x$ are continuous everywhere because they satisfy all three conditions of continuity at every point $a$.