Continuous Functions
Key Questions

You can prove that a function
#f(x)# is continuous at#a# by verifying that#lim_{x to a}f(x)=f(a)# . 
No, it is not continuous at
#x=3# since it is not defined there (zero denominator). 
Answer:
We may also state two alternative definitions of continuous functions, using either the sequential criterion or else using topology and open sets.
Explanation:
Alternative definition number 1
Let#f: X >Y# be a function and let#(x_n)# be a sequence in X converging to an element x in X, ie#lim(x_n)=x in X#
Then f is continuous at x iff and only if the sequence of function values converge to the image of x undr f, ie#iff lim (f(x_n))=f(x) in Y#
Alternative definition number 2
Let#f: X >Y# be a function. Then f is continuous if the inverse image maps open subsets of Y into open subsets in X.
ie,#AAA_(open)subeY =>f^(1)(A) # is open in X
Questions
Limits

Introduction to Limits

Determining One Sided Limits

Determining When a Limit does not Exist

Determining Limits Algebraically

Infinite Limits and Vertical Asymptotes

Limits at Infinity and Horizontal Asymptotes

Definition of Continuity at a Point

Classifying Topics of Discontinuity (removable vs. nonremovable)

Determining Limits Graphically

Formal Definition of a Limit at a Point

Continuous Functions

Intemediate Value Theorem

Limits for The Squeeze Theorem