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1.7. LIMITS, CONTINUITY, AND DIFFERENTIABILITY
• A function f is continuous at x = a whenever f (a) is defined, f has a limit as x → a,
and the value of the limit and the value of the function agree. This guarantees that
there is not a hole or jump in the graph of f at x = a.
• A function f is differentiable at x = a whenever f ′ (a) exists, which means that f has a
tangent line at (a, f (a)) and thus f is locally linear at the value x = a. Informally, this
means that the function looks like a line when viewed up close at (a, f (a)) and that
there is not a corner point or cusp at (a, f (a)).
• Of the three conditions discussed in this section (having a limit at x = a, being
continuous at x = a, and being differentiable at x = a), the strongest condition is
being differentiable, and the next strongest is being continuous. In particular, if f is
differentiable at x = a, then f is also continuous at x = a, and if f is continuous at
x = a, then f has a limit at x = a.
Exercises
1. Consider the graph of the function y = p(x) that is provided in Figure 1.42. Assume
that each portion of the graph of p is a straight line, as pictured.
-3
3
-3
3
p
-3
3
-3
3
Figure 1.42: At left, the piecewise linear function y = p(x). At right, axes for plotting
y = p ′ (x).
(a) State all values of a for which lim x→a p(x) does not exist.
(b) State all values of a for which p is not continuous at a.
(c) State all values of a for which p is not differentiable at x = a.
(d) On the axes provided in Figure 1.42, sketch an accurate graph of y = p ′ (x).
2. For each of the following prompts, give an example of a function that satisfies the
stated criteria. A formula or a graph, with reasoning, is sufficient for each. If no such
example is possible, explain why.
1.7. LIMITS, CONTINUITY, AND DIFFERENTIABILITY
• A function f is continuous at x = a whenever f (a) is defined, f has a limit as x → a,
and the value of the limit and the value of the function agree. This guarantees that
there is not a hole or jump in the graph of f at x = a.
• A function f is differentiable at x = a whenever f ′ (a) exists, which means that f has a
tangent line at (a, f (a)) and thus f is locally linear at the value x = a. Informally, this
means that the function looks like a line when viewed up close at (a, f (a)) and that
there is not a corner point or cusp at (a, f (a)).
• Of the three conditions discussed in this section (having a limit at x = a, being
continuous at x = a, and being differentiable at x = a), the strongest condition is
being differentiable, and the next strongest is being continuous. In particular, if f is
differentiable at x = a, then f is also continuous at x = a, and if f is continuous at
x = a, then f has a limit at x = a.
Exercises
1. Consider the graph of the function y = p(x) that is provided in Figure 1.42. Assume
that each portion of the graph of p is a straight line, as pictured.
-3
3
-3
3
p
-3
3
-3
3
Figure 1.42: At left, the piecewise linear function y = p(x). At right, axes for plotting
y = p ′ (x).
(a) State all values of a for which lim x→a p(x) does not exist.
(b) State all values of a for which p is not continuous at a.
(c) State all values of a for which p is not differentiable at x = a.
(d) On the axes provided in Figure 1.42, sketch an accurate graph of y = p ′ (x).
2. For each of the following prompts, give an example of a function that satisfies the
stated criteria. A formula or a graph, with reasoning, is sufficient for each. If no such
example is possible, explain why.