SINGLE VARIABLE CALCULUS. PART I: DIFFERENTIATION. PART A: DEFINITION AND BASIC RULES. UNIT 0: LIMITS REVIEW. PART 2: Continuity

 SINGLE VARIABLE CALCULUS

PART I: DIFFERENTIATION

PART A: DEFINITION AND BASIC RULES

UNIT 0: LIMITS REVIEW

PART 2: Continuity

I. Definition of continuity at a point

    We say that a function $f$  is continuous at a point $x = a$ if

    In particular, if $f(a)$ or the limit of $f(x)$ when $x$ approaches $a$ fails, the function is discontinuous at $x = a$.

    We say that the function $f$ is right-continuous at point $x = a$ if

    Familiarly, we have the left-continuous at point $x = a$ :

II. Types of Discontinuities:

    There are only two types:

        1. If there are a left-hand limit and right-hand limit at point $x = a$, but they are not equal, then we say that there is a jump discontinuity at point $x = a$.

        2. If the overall limit of $f(x)$ when x approaches a exists, but $f(a)$ does not equal to that limit, then there is a removable discontinuity at $x = a$.

III. Definition of Continuous functions.

    The function $f(x)$ is continuous whether, for every point c in its domain, the function is continuous at every $x = c$.

IV. Limit laws and Continuity

    Given functions $f$ and $g$ are continuous everywhere, then:

• $f + g$ is continuous everywhere.
• $f - g$  is continuous everywhere.
• $f.g$ is continuous everywhere.
• $f/g$ is continuous where it is defined.

 V. Immediate Value Theorem

        If a function f which is continuous on an interval [a,b] and the value M lies between $f(a)$ and $f(b)$, then there is at least one $x = c$ between $a$ and $b$ that $f(c) = M$.

    ( A function $f$ is continuous on a closed interval [a,b] if it is right-continuous at $x = a$ or left-continuous at $x = b$ or continuous at every point on [a,b] 

 Appendix: Basic Continuous Functions.

• all polynomials
• |x|
• cosx and sinx
• exponential functions a^x with base a>0
• x^1/3
• $√x$ for $x>0$
• tanx for all $x$, where it is defined.
• logarithmic functions $$log_ax$$ with base $a > 0$ and $x>0$.