Q.Check the points where the constant function $f(x) = k$ is continuous.

The key idea is that a constant function does not change — its value is the same at every point, so the limit at any point equals the function value there.

Step 1: For any $c$ in the domain (all real numbers), the function value is $f(c) = k$.

Step 2: The limit as $x$ approaches $c$ is $\lim_{x \to c} f(x) = \lim_{x \to c} k = k$. …

The Core Idea: Continuity at a Point

Before we check any specific point, let's recall what continuity means at a single point $x = a$. A function $f$ is continuous at $a$ if three things hold:

1. $f(a)$ is defined (the function has a value at $a$).
2. $\lim_{x \to a} f(x)$ exists.
3. $\lim_{x \to a} f(x) = f(a)$.

The third condition is the one that usually causes trouble. For most functions, you have to check whether the left-hand limit and right-hand limit match, and then whether they match the function's value.

But here, the function is $f(x) = k$. It's the simplest possible function — it outputs the same number $k$ no matter what $x$ you plug in. This simplicity is the key.

Step-by-Step Reasoning

Let's pick an arbitrary point $x = a$ on the real number line. We'll check the three conditions.

1. Is $f(a)$ defined?

Yes. $f(a) = k$. The function is defined for every real $a$, so this condition is satisfied for any $a$ you choose.

2. Does $\lim_{x \to a} f(x)$ exist?

We need to see what happens to $f(x)$ as $x$ gets arbitrarily close to $a$, from either side.

Since $f(x) = k$ for every $x$, as $x$ approaches $a$ from the left ($x \to a^-$), the function value is always $k$. So the left-hand limit is:

Similarly, as $x$ approaches $a$ from the right ($x \to a^+$), the function value is still $k$. So the right-hand limit is:

Both one-sided limits exist and are equal to $k$. Therefore, the two-sided limit exists and is also $k$:

3. Does $\lim_{x \to a} f(x) = f(a)$? …

Method: Proving a Function Is Continuous at an Arbitrary Point

This method proves continuity not at one numerical point but at every point of the domain, by picking an arbitrary point, applying the continuity conditions symbolically, and noting the argument never depended on which point was chosen.

Steps

Step 1: Let the point be an arbitrary, unspecified value

Instead of substituting a number, let the test point be a general symbol (commonly $a$ or $c$), standing for "any real number."

Step 2: Evaluate the function value symbolically

Write $f(a)$ using the function's rule, treating $a$ as fixed but unspecified.

Step 3: Compute the limit symbolically

$\lim_{x \to a} f(x)$ …