Limits – Continuity and Intermediate Value Theorem A continuous function, is one that has no gaps or holes along a given domain.

Limits – Continuity and Intermediate Value Theorem A continuous function, is one that has no gaps or holes along a given domain.    

Limits – Continuity and Intermediate Value Theorem A continuous function, is one that has no gaps or holes along a given domain.   This function has no gaps, or holes, and continues on in both directions without bound.  

Limits – Continuity and Intermediate Value Theorem A continuous function, is one that has no gaps or holes along a given domain.   This function has no gaps, or holes, and continues on in both directions without bound. I could trace this graph between any ( a , b ) interval without lifting the pencil off of the paper…  

Limits – Continuity and Intermediate Value Theorem A continuous function, is one that has no gaps or holes along a given domain.     This function is continuous but only within a restricted domain or a closed interval. - 2 2  

Limits – Continuity and Intermediate Value Theorem   - if any of these conditions are false, then the function is not continuous at “c”    

Limits – Continuity and Intermediate Value Theorem   - if any of these conditions are false, then the function is not continuous at “c”      

Limits – Continuity and Intermediate Value Theorem   - if any of these conditions are false, then the function is not continuous at “c”       Functions that are not continuous usually have domain values that are undefined or the function is a Sigma or Piecewise function.

Limits – Continuity and Intermediate Value Theorem   - if any of these conditions are false, then the function is not continuous at “c”       Functions that are not continuous usually have domain values that are undefined or the function is a Sigma or Piecewise function. The Intermediate Value Theorem merely states that if I have a continuous function on a closed interval [ a , b ], if I choose a value within that given interval , the function will be continuous at that point.

Limits – Continuity and Intermediate Value Theorem Sigma or Piecewise functions have different definitions on specific intervals.

Limits – Continuity and Intermediate Value Theorem Sigma or Piecewise functions have different definitions on specific intervals. EXAMPLE :      

Limits – Continuity and Intermediate Value Theorem Sigma or Piecewise functions have different definitions on specific intervals. EXAMPLE :   As you can see, ƒ(x) has two definitions…    

Limits – Continuity and Intermediate Value Theorem Sigma or Piecewise functions have different definitions on specific intervals. EXAMPLE :   As you can see, ƒ(x) has two definitions…     Here is what the graph of this function would look like.

Limits – Continuity and Intermediate Value Theorem Another way to describe if a function is continuous : if the limit as “x” approaches “c” from the right of “c”= the limit as “x” approached “c” from the left of “c” , then the function is continuous at “c”  

Limits – Continuity and Intermediate Value Theorem Another way to describe if a function is continuous : if the limit as “x” approaches “c” from the right of “c”= the limit as “x” approached “c” from the left of “c” , then the function is continuous at “c”   We have explored limits approaching a value from the left/right to check if they are equal using a table so another example is not necessary…