if and only
if f(x) is continuous at x = c.
If a function is continuous at every point in an interval, then we say that the function is continuous in that interval. And if a function is continuous in any interval, then we simply call it a continuous function.
Calculus is fundamentally about functions that are continuous at every point in their domains. Prime examples of continuous functions are polynomials (Lesson 2).
Problem 1.
a) Prove that this polynomial,
2x² − 3x + 5,
a) is continuous at x = 1.
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Do the problem yourself first!
We must apply the definition of continuous at a point, Definition 3.1. That is, we must show that when x approaches 1 as a limit, f(x) approaches f(1), which is 4.
And according to the Theorems of Lesson 2, that is true.
f(x) therefore is continuous at x = 1.
b) Can you think of any value of x where that polynomial -- or any
b) polynomial -- would not be continuous?
You should not be able to. Polynomials are continuous everywhere. As x approaches any limit c, any polynomial
P(x) approaches P(c). (Lesson 2)
In addition to polynomials, the following functions also are continuous at every point in their domains.
Rational functions
Root functions
Trigonometric functions
Inverse trigonometric functions
Logarithmic functions
Exponential functions
These are the functions that one encounters throughout calculus. To evaluate the limit of any one of these as x approaches a value, simply evaluate the function at that value.