Understanding Differentiability and Continuity in Functions
Explore the connection between differentiability and continuity in functions. Learn how continuous functions may not necessarily be differentiable and vice versa. Dive into the concept that differentiable functions are continuous, and discover the implications of locally linear functions.
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Presentation Transcript
Differentiable functions are Continuous Connecting Differentiability and Continuity
Differentiability and Continuity Continuous functions are not necessarily differentiable. For instance, start with = 2 ( ) f x 4 x ( 2,0) (2,0) (0, 4)
Differentiability and Continuity Continuous functions are not necessarily differentiable. . . . Now take absolute values = 2 ( ) f x 4 x ( 2,0) (2,0) (0, 4)
Differentiability and Continuity Continuous functions are not necessarily differentiable. . . . Now take absolute values = 2 ( ) f x 4 x (0,4) ( 2,0) (2,0) (0, 4)
Differentiability and Continuity Continuous functions are not necessarily differentiable. (E.g. ) A function is differentiable if we see a straight line when we zoom in sufficiently far. = 2 ( ) f x 4 x
Differentiability and Continuity Continuous functions are not necessarily differentiable. (E.g. ) A function is differentiable if we see a straight line when we zoom in sufficiently far. Our intuition thus tells us that locally linear functions cannot have breaks in the graph. But how do we prove this? = 2 ( ) f x 4 x
Differentiable Functions are Continuous Suppose that f is differentiable at x = a. Then ( ) f x ( ) f a ( ) = lim x . f a x a a In order to show that f is continuous at x = a, we have to show that = lim ( ) x ( ) f a af x
Wed like to break Differentiable Functions are Continuous up this limit. Can we do it? What do we need to know? Suppose that f is differentiable at x = a. Then we know the limit of the difference quotient exists and is equal to . ( ) f a ( ) f x ( ) f a ( ) = lim x f a x a a ( ) = + lim ( ) x a lim x ( ) f x ( ) f a ( ) f a f x a ( ) f x ( ) f a = x a + lim x ( ) ( ) f a x a a ( ) f x ( ) f a = x a + lim x lim( x ) lim ( ) x a f a x a a a = + = ( )(0) f a ( ) f a ( ) f a
Differentiable Functions are Continuous In the end, this tells us that: = lim ( ) x ( ) f a af x Which is what it means to say that f is continuous at a ! So if f is differentiable at x = a, then f must also be continuous at x = a
