Understanding Continuous and Discontinuous Functions

Continuous Functions

Intuitive Idea:
A continuous function is one whose graph can be drawn without lifting your pen from the paper. There are no breaks, jumps, or holes in the graph.

Formal Definition at a Point:
A function $f(x)$ is said to be *continuous at a point$x = c$ if it meets the following three conditions:

$f(c)$ is defined (i.e., $c$ is in the domain of $f$ ).
The limit of $f(x)$ as $x$ approaches $c$ exists (i.e., $\lim_{x \to c} f(x)$ exists). This means the limit from the left equals the limit from the right ($\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$).
The limit of $f(x)$ as $x$ approaches $c$ is equal to the function’s value at $x = c$ (i.e., $\lim_{x \to c} f(x) = f(c)$ ).

Continuous on an Interval:
A function is continuous on an interval if it is continuous at every point within that interval.

Examples of Continuous Functions:
-Polynomial functions: e.g., $f(x) = x^2 + 3x - 2$ . These are continuous everywhere.
-Trigonometric functions: $f(x) = \sin(x)$ and $f(x) = \cos(x)$ are continuous everywhere. Functions like $\tan(x)$ , $\sec(x)$ , $\csc(x)$ , and $\cot(x)$ are continuous on their respective domains (they have discontinuities where their denominators are zero).
-Exponential functions: e.g., $f(x) = e^x$ or $f(x) = 2^x$ . These are continuous everywhere.
-Logarithmic functions: e.g., $f(x) = \ln(x)$ or $f(x) = \log_{10}(x)$ . These are continuous on their domain (for $x > 0$ ).

Discontinuous Functions

Intuitive Idea:
A discontinuous function is one whose graph has one or more breaks, jumps, or holes. You would have to lift your pen from the paper to draw its graph.

Reasons for Discontinuity:
A function $f(x)$ is discontinuous at a point $x = c$ if any of the conditions for continuity are not met. For instance:
The function is not defined at $x = c$ .
The limit of $f(x)$ as $x$ approaches $c$ does not exist (e.g., the left-hand limit and right-hand limit are different).
The limit of $f(x)$ as $x$ approaches $c$ exists, and $f(c)$ is defined, but the limit is not equal to $f(c)$ .

Importance:
The concept of continuity is crucial in calculus. For example:

Differentiability implies continuity: If a function is differentiable at a point, it must be continuous at that point. However, a continuous function is not necessarily differentiable (e.g., $f(x) = |x|$ is continuous at $x=0$ but not differentiable there).
Intermediate Value Theorem: If a function is continuous on a closed interval $[a, b]$ , then it takes on every value between $f(a)$ and $f(b)$ .
Extreme Value Theorem: If a function is continuous on a closed interval $[a, b]$ , then it attains an absolute maximum value and an absolute minimum value on that interval.
Integrability: Continuous functions on a closed interval are integrable on that interval.