Continuity

Definition of Continuity at a Point

Let $f$ be a real function defined on a subset of $\mathbb{R}$, and let $c$ be a point in the domain of $f$.

Then $f$ is continuous at $c$ if:

More elaborately, this means all three of the following conditions hold:

1. $f(c)$ is defined — the function has a value at $x = c$.
2. $\displaystyle \lim_{x \to c} f(x)$ exists — both the left-hand limit $\displaystyle \lim_{x \to c^-} f(x)$ and the right-hand limit $\displaystyle \lim_{x \to c^+} f(x)$ exist and are equal.
3. The limit equals the function value — $\displaystyle \lim_{x \to c} f(x) = f(c)$.

If any of these fails, $f$ is discontinuous at $c$, and $c$ is called a point of discontinuity.

Intuition

A function is continuous at a point if you can draw its graph without lifting your pen near that point — there is no sudden jump, hole, or break.

Tiny Concrete Example

Let $f(x) = 2x + 3$. At $x = 1$: …

Definition of Continuity (Global)

A real function $f$ is said to be continuous if it is continuous at every point in its domain.

This means:

• For every point $c$ in the domain of $f$, the condition

must hold.

• Equivalently, at each $c$, the left-hand limit, right-hand limit, and the value of the function must all exist and be equal.

Special cases for closed intervals $[a, b]$:

• At the left endpoint $a$, continuity means $\displaystyle \lim_{x \to a^+} f(x) = f(a)$.
• At the right endpoint $b$, continuity means $\displaystyle \lim_{x \to b^-} f(x) = f(b)$.
• Limits from outside the interval do not apply.

If the domain is a single point, the function is automatically continuous there.

Intuition

A function is continuous if you can draw its entire graph without lifting your pen — no breaks, jumps, or holes anywhere in the domain.

--- …

Fig 5.1 is the first visual the textbook uses to introduce the idea of continuity. It plots a simple step function on the standard $X'X$/$Y'Y$ Cartesian axes, with the origin labelled $O$. The function is defined piecewise:

The graph shows two horizontal line segments. For $x \le 0$, there is an indigo segment at height $y = 1$ that ends with a solid dot at the point $(0,1)$ — this dot means the function actually takes the value $1$ at $x = 0$. For $x > 0$, a separate indigo segment sits at height $y = 2$, but it begins with an open circle at $(0,2)$ — that open circle tells you the function does not take the value $2$ at $x = 0$; the point $(0,2)$ is not on the graph. The two levels do not meet; there is a clear vertical gap, a jump, at $x = 0$.

The physical idea is immediate: as you move along the $x$-axis from left to right, the function's value stays at $1$ until you reach $0$, then it suddenly leaps to $2$. If you try to draw this graph with a pen, you must lift the pen off the paper at $x = 0$ to jump from the lower segment to the upper one — you cannot trace the whole curve in one continuous stroke. That lifting of the pen is the intuitive signal that the function is not continuous at $x = 0$.

The textbook uses this figure to ground the formal definition of continuity. It observes that the left-hand limit of $f$ at $0$ is $1$ (because points like $-0.1, -0.01, -0.001$ all give $f(x) = 1$), while the right-hand limit is $2$ (points like $0.1, 0.01, 0.001$ give $f(x) = 2$). Since these two one-sided limits are different, the ordinary limit $\lim_{x \to 0} f(x)$ does not exist. The value of the function at $x = 0$ is $f(0) = 1$, which happens to equal the left-hand limit but not the right-hand limit. The formal condition for continuity at a point $c$ is:

Equivalently, all three numbers — left-hand limit, right-hand limit, and the function value — must exist and be equal. Here, the left-hand limit ($1$) and right-hand limit ($2$) are unequal, so the limit does not exist, and the equality fails. Hence $f$ is discontinuous at $x = 0$, and $x = 0$ is called a point of discontinuity. …

Fig 5.2 is a simple but powerful picture of a function that is defined at every point yet still fails to be continuous. The axes are the standard Cartesian axes with origin O. A single horizontal line at $y = 1$ stretches across the entire width of the graph, with arrowheads at both ends to show it continues indefinitely. But this line is broken: there is an open circle (a hole) at the point $(0, 1)$. Separately, a solid dot sits directly above that hole at $(0, 2)$.

The function being graphed is

The horizontal line at $y = 1$ represents all points where $x \neq 0$ — every real number except zero gives an output of 1. The open circle at $(0, 1)$ tells you that the function does not take the value 1 at $x = 0$, even though the points arbitrarily close to 0 from both sides do. The solid dot at $(0, 2)$ is the actual value of the function at $x = 0$: $f(0) = 2$.

The physical idea is that the left-hand limit and the right-hand limit both exist and are equal — both are 1 — but the function's value at the point is different. The common limit exists ($\lim_{x \to 0} f(x) = 1$), yet $f(0) = 2 \neq 1$. This is a removable discontinuity: you could "fix" the function by redefining $f(0)$ to be 1, and the graph would become a continuous horizontal line. The hole-and-dot picture makes this instantly visible — the function jumps up to 2 at exactly one isolated point.

The key formula the textbook develops with this figure is the formal definition of continuity at a point: …

Fig 5.3 is the graph of the reciprocal function $f(x) = \frac{1}{x}$, drawn for $x \neq 0$. The plot has two separate hyperbolic branches, one in the first quadrant and one in the third quadrant, each shown in indigo. The horizontal axis is the $x$-axis and the vertical axis is the $y$-axis. On the $y$-axis, ticks are marked at $1, 2, 3$ and $-1, -2, -3$.

The first-quadrant branch passes through the points $\left(\frac13, 3\right)$, $\left(\frac12, 2\right)$, $(1, 1)$, and $(2, \frac12)$. As $x$ increases beyond $2$, the curve falls toward the $x$-axis without ever touching it. As $x$ approaches $0$ from the right, the curve rises steeply toward the $y$-axis, again without touching it. The third-quadrant branch is the mirror image through the origin: it passes through $\left(-2, -\frac12\right)$, $(-1, -1)$, $\left(-\frac12, -2\right)$, and $\left(-\frac13, -3\right)$. As $x$ moves leftward from $-2$, the curve approaches the $x$-axis from below; as $x$ approaches $0$ from the left, the curve plunges downward toward the $y$-axis.

The central idea this figure teaches is the behaviour of a function near a point where it is not defined — here, $x = 0$. The textbook uses the graph to illustrate the concepts of infinite limits. As $x$ gets closer to $0$ from the right, the value of $f(x)$ grows without bound. In symbols:

As $x$ approaches $0$ from the left, the value becomes arbitrarily negative:

The figure also makes clear that the $x$-axis ($y = 0$) and the $y$-axis ($x = 0$) are asymptotes: the curve gets arbitrarily close to each axis but never crosses it. This is the geometric reason why $f(x) = \frac{1}{x}$ is continuous at every point in its domain (all real numbers except $0$), but has a vertical asymptote at $x = 0$ where it is not defined and therefore not continuous.

| $x$ | $1$ | $0.3$ | $0.2$ | $0.1=10^{-1}$ | $0.01=10^{-2}$ | $0.001=10^{-3}$ | $10^{-n}$ |

|---|---|---|---|---|---|---|---| …

| $x$ | $-1$ | $-0.3$ | $-0.2$ | $-10^{-1}$ | $-10^{-2}$ | $-10^{-3}$ | $-10^{-n}$ |

|---|---|---|---|---|---|---|---| …