Continuous Functions - Concept
In order for a function to have continuity at a certain point, several conditions including (1) the existence of the point in the domain and (2) the existence of a two-sided limit as the point approaches the limit must be met. If functions are continuous at every point in their domain, they we call these functions continuous functions. Examples of continuous functions are power functions, exponential functions and logarithmic functions.
Okay we defined the idea of Continuity before a function f is continuous at a point x=a if the limit as x approaches a of f of x equals f of a but we have a further definition. If a function f is continuous at every point in its domain then we call f a continuous function. So when we say continuous function we mean it's continuous at every point in its domain, let's take a look at some examples. So all power functions are continuous for example f of x equals x cubed that's a power function it's defined for all real numbers so it's continuous everywhere it's defined, it's continuous I'll abbreviate that CONT on the interval from negative infinity to infinity, g of x equals x to the -2 that's also a power function it's continuous everywhere it's defined, right? This is the same by the way 1 over x squared. This will be continuous for all x except 0 continuous for x not 0.
And how about one more uh let's call it h of x. h of x equals x to the one half this is the same as the square root of x but it is a power function this is only defined for x greater than or equal 0. That's a power function so it's continuous every where it's defined it's continuous for x greater than or equal to 0. So remember all power functions are continuous. Then all exponential functions are continuous examples f of x equals 3 to the x g of x equals 10 to the x, h of x equals e to the x. All of these functions all exponential functions are continuous everywhere. They're defined for all real numbers so all of them are continuous from negative infinity to infinity.
And one more class of functions "Log Arithmetic Functions," are continuous functions. Now they're only defined for positive numbers but still they're continuous everywhere they're defined so they're continuous and examples would be f of x equals the common log, log base 10 is the common log where it's often written this way log x that's an example of a log function g of x equals log base one third of x another log function. And hopefully you remember log base e of x, that's the natural log of x. All three of these guys are defined only for positive values of x and so that's where they're continuous. They're continuous for x greater than zero. So once again a continuous function is a function that's continuous at every point in its domain.