Question Statement

Consider the function: $f (x) = x^{2} - 5 x + 6$

Determine whether this function has any points of discontinuity.

Background and Explanation

Continuity is a fundamental property in calculus describing functions without breaks or jumps. Polynomial functions represent one of the most well-behaved classes of functions, possessing continuity across their entire domain.

Solution

We begin by examining the given function: $f (x) = x^{2} - 5 x + 6$

This expression represents a polynomial function—specifically, a quadratic polynomial (degree 2) formed by the terms $x^{2}$ , $- 5 x$ , and $+ 6$ .

The function $f (x)$ is a polynomial function, and polynomial functions are continuous for all real numbers. This property arises because polynomials are constructed from power functions and constants using only addition, subtraction, and multiplication, all of which preserve continuity.

Therefore, since $f (x)$ is continuous everywhere on the real number line, we conclude that:

$f (x)$ has no point of discontinuity.

Key Formulas or Methods Used

Polynomial Continuity Theorem: All polynomial functions $P (x)$ are continuous on the entire real line $(- \infty, \infty)$
Polynomial Identification: Recognizing algebraic expressions of the form $a_{n} x^{n} + \dots + a_{1} x + a_{0}$ as polynomials

Summary of Steps

Identify the given function $f (x) = x^{2} - 5 x + 6$ as a polynomial function
Apply the theorem stating that polynomial functions are continuous for all real numbers
Conclude that the function has no points of discontinuity

Question Statement

Consider the function: $f (x) = x^{2} - 5 x + 6$

Determine whether this function has any points of discontinuity.

Background and Explanation

Solution

We begin by examining the given function: $f (x) = x^{2} - 5 x + 6$

This expression represents a polynomial function—specifically, a quadratic polynomial (degree 2) formed by the terms $x^{2}$ , $- 5 x$ , and $+ 6$ .

Therefore, since $f (x)$ is continuous everywhere on the real number line, we conclude that:

$f (x)$ has no point of discontinuity.

Key Formulas or Methods Used

Polynomial Continuity Theorem: All polynomial functions $P (x)$ are continuous on the entire real line $(- \infty, \infty)$
Polynomial Identification: Recognizing algebraic expressions of the form $a_{n} x^{n} + \dots + a_{1} x + a_{0}$ as polynomials

Summary of Steps

Identify the given function $f (x) = x^{2} - 5 x + 6$ as a polynomial function
Apply the theorem stating that polynomial functions are continuous for all real numbers
Conclude that the function has no points of discontinuity

2.2 Q-1

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

2.2 Q-1

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps