An introductory course that explores the fundamental concepts and techniques of mathematical analysis.
A function connects every input from one set (like real numbers) to exactly one output in another set. Think of it as a math machine: put in a number, get out a number!
A function is continuous at a point if its output doesn't suddenly jump or break there. You can draw its graph without lifting your pencil.
A function \( f \) is continuous at \( x = a \) if: \[ \lim_{x \to a} f(x) = f(a) \] This means the value you get by plugging in numbers close to \( a \) is the same as just plugging in \( a \).
Continuous functions model real-world processes smoothly, like the growth of a plant or the change in temperature over time.
Imagine walking along a path. If the path is continuous, you never have to jump over holes!
The function \( f(x) = x^2 \) is continuous everywhere.
The function \( f(x) = \frac{1}{x} \) is not continuous at \( x = 0 \), because you can't divide by zero.
Functions assign each input to an output, and continuity means their graphs are smooth with no jumps.