Q.Show that the derivative of a constant is zero and the derivative of $ax$ with respect to $x$ is $a$.

Continuity of a Function

The Intuition: Drawing Without Lifting the Pen

Imagine you are drawing the graph of a function on a piece of paper. If you can trace the entire curve without lifting your pen from the paper, the function is continuous. Every time you have to lift the pen — because the graph jumps, breaks, or has a hole — the function is discontinuous at that point.

That is the visual idea. A continuous function has no sudden leaps, no gaps, no punctures. Its output changes smoothly as its input changes.

Consider a simple example: $f(x) = x^2$. As $x$ moves from 1 to 2, the output moves from 1 to 4, passing through every value in between. No jump, no missing point. You can draw it in one stroke.

Now contrast that with a function like:

At $x=1$, the graph has a single isolated point at height 5, while the rest of the curve approaches height 1. To draw this, you would trace the parabola, then lift your pen to place a dot at $(1,5)$. That lift is the discontinuity.

The Problem with Intuition Alone

"Drawing without lifting the pen" works for simple functions, but it fails for strange ones. Some functions are continuous yet impossible to draw (like the Weierstrass function, which is continuous but has no smooth tangent anywhere). More practically, the pen-lifting test is not a mathematical definition — it cannot tell you exactly what "no break" means at a single point.

We need a precise, point-by-point definition.

The Precise Definition: The Three-Part Test

A function $f(x)$ is said to be continuous at a point $x = a$ if and only if all three of the following conditions hold:

1. $f(a)$ is defined. The function must have a value at $x = a$. No holes.
2. $\lim_{x \to a} f(x)$ exists. As $x$ gets arbitrarily close to $a$ from either side, the function's values must approach a single finite number.
3. $\lim_{x \to a} f(x) = f(a)$. The limit must equal the actual function value. The point must sit exactly where the surrounding curve is heading.

If any one of these fails, the function is discontinuous at $x = a$.

Why the Limit Matters

The limit captures the trend of the function near $a$, ignoring what happens exactly at $a$. Continuity demands that this trend matches the actual point. This is why the earlier piecewise function fails: the limit as $x \to 1$ is $1$, but $f(1) = 5$, so condition 3 is violated.

A Worked Example

Test whether $f(x) = \frac{x^2 - 1}{x - 1}$ is continuous at $x = 1$.

Step 1: Is $f(1)$ defined?

No. The denominator becomes zero, so $f(1)$ is undefined. Condition 1 fails immediately. The function is discontinuous at $x = 1$ — it has a hole there.

Even though $\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1} (x+1) = 2$, the function never actually takes the value 2 at $x=1$. The hole is a discontinuity.

Continuity on an Interval

A function is continuous on an interval (like $(a,b)$ or $[a,b]$) if it is continuous at every point in that interval. For a closed interval $[a,b]$, we only require one-sided continuity at the endpoints: from the right at $a$, and from the left at $b$.

The Big Picture

Continuity is the mathematical way of saying "no surprises." It guarantees that small changes in input produce small changes in output. This property is the foundation for everything that follows in calculus: the Intermediate Value Theorem, differentiability (every differentiable function is continuous), and the ability to evaluate limits by direct substitution for continuous functions.