1.7. LIMITS, CONTINUITY, AND DIFFERENTIABILITY
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f
Figure 1.39: The graph of y = f (x) for Activity 1.19.
Being differentiable at a point
We recall that a function f is said to be differentiable at x = a whenever f ′ (a) exists.
Moreover, for f ′ (a) to exist, we know that the function y = f (x) must have a tangent line
at the point (a, f (a)), since f ′ (a) is precisely the slope of this line. In order to even ask
if f has a tangent line at (a, f (a)), it is necessary that f be continuous at x = a: if f
fails to have a limit at x = a, if f (a) is not defined, or if f (a) does not equal the value of
lim x→a f (x), then it doesn’t even make sense to talk about a tangent line to the curve at
this point.
Indeed, it can be proved formally that if a function f is differentiable at x = a, then it
must be continuous at x = a. So, if f is not continuous at x = a, then it is automatically
the case that f is not differentiable there. For example, in Figure 1.38 from our early
discussion of continuity, both f and g fail to be differentiable at x = 1 because neither
function is continuous at x = 1. But can a function fail to be differentiable at a point
where the function is continuous?
In Figure 1.40, we revisit the situation where a function has a sharp corner at a point,
something we encountered several times in Section 1.4. For the pictured function f , we
observe that f is clearly continuous at a = 1, since lim x→1 f (x) = 1 = f (1).
But the function f in Figure 1.40 is not differentiable at a = 1 because f ′ (1) fails to
exist. One way to see this is to observe that f ′ (x) = −1 for every value of x that is less
than 1, while f ′ (x) = +1 for every value of x that is greater than 1. That makes it seem that
either +1 or −1 would be equally good candidates for the value of the derivative at x = 1.
Alternately, we could use the limit definition of the derivative to attempt to compute f ′ (1),
and discover that the derivative does not exist. A similar problem will be investigated in
Activity 1.20. Finally, we can also see visually that the function f in Figure 1.40 does not
have a tangent line. When we zoom in on (1, 1) on the graph of f , no matter how closely
we examine the function, it will always look like a “V”, and never like a single line, which
71
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1 2 3
-3
-2
-1
1
2
3
f
Figure 1.39: The graph of y = f (x) for Activity 1.19.
Being differentiable at a point
We recall that a function f is said to be differentiable at x = a whenever f ′ (a) exists.
Moreover, for f ′ (a) to exist, we know that the function y = f (x) must have a tangent line
at the point (a, f (a)), since f ′ (a) is precisely the slope of this line. In order to even ask
if f has a tangent line at (a, f (a)), it is necessary that f be continuous at x = a: if f
fails to have a limit at x = a, if f (a) is not defined, or if f (a) does not equal the value of
lim x→a f (x), then it doesn’t even make sense to talk about a tangent line to the curve at
this point.
Indeed, it can be proved formally that if a function f is differentiable at x = a, then it
must be continuous at x = a. So, if f is not continuous at x = a, then it is automatically
the case that f is not differentiable there. For example, in Figure 1.38 from our early
discussion of continuity, both f and g fail to be differentiable at x = 1 because neither
function is continuous at x = 1. But can a function fail to be differentiable at a point
where the function is continuous?
In Figure 1.40, we revisit the situation where a function has a sharp corner at a point,
something we encountered several times in Section 1.4. For the pictured function f , we
observe that f is clearly continuous at a = 1, since lim x→1 f (x) = 1 = f (1).
But the function f in Figure 1.40 is not differentiable at a = 1 because f ′ (1) fails to
exist. One way to see this is to observe that f ′ (x) = −1 for every value of x that is less
than 1, while f ′ (x) = +1 for every value of x that is greater than 1. That makes it seem that
either +1 or −1 would be equally good candidates for the value of the derivative at x = 1.
Alternately, we could use the limit definition of the derivative to attempt to compute f ′ (1),
and discover that the derivative does not exist. A similar problem will be investigated in
Activity 1.20. Finally, we can also see visually that the function f in Figure 1.40 does not
have a tangent line. When we zoom in on (1, 1) on the graph of f , no matter how closely
we examine the function, it will always look like a “V”, and never like a single line, which