Reading a Graph Near a Marked Point
A graph can show what a function does at a particular input and what it does nearby. Those are different pieces of information. To test continuity at a marked input \(x=a\), you need both: the actual function value \(f(a)\), if it is defined, and the value the graph approaches as \(x\) moves toward \(a\).
In a graph, a filled dot at \(x=a\) indicates the function’s value there. An open circle indicates a point the drawn curve approaches but does not include as a function value at that input. If the graph has a filled dot and an open circle at the same input but different heights, the filled dot gives \(f(a)\); the open circle can indicate the value approached along the curve.
These symbols are clues, not a complete test. In particular, an open circle alone does not tell you whether the left and right sides approach the same height. Inspect both sides of the marked input.
The Three Checks
For a function to be continuous at \(x=a\), the function must be defined at \(a\), the limit as \(x\) approaches \(a\) must exist as a finite value, and that limit must equal \(f(a)\). These are the three conditions from the earlier material on continuity at a point, now read from a graph.
For an interior point of the graph’s domain, the two-sided limit exists precisely when the left-hand and right-hand limits both exist as finite values and are equal. In symbols, if these one-sided limits are \(L_-\) and \(L_+\), then the required condition is \(L_-=L_+\). If they are unequal, there is no single value approached from both sides.
The three checks separate questions that can look similar in a picture. A function may have a value at \(a\) but fail to approach it. It may approach one value but have no function value there. Or it may approach one value and be defined there, but at a different height.
Why the Graph Test Matches the Definition
Continuity at \(a\) means that \(f(a)\) exists and that \(\lim_{x\to a}f(x)=f(a)\). A two-sided limit at an interior point exists when both one-sided limits exist and agree. Combining those facts gives the three graph checks.
To see why the one-sided test captures the two-sided limit, suppose the left- and right-hand limits both equal the same finite number \(L\). Given any tolerance \(\varepsilon>0\), the left-hand limit supplies a distance \(\delta_->0\) that keeps the function within \(\varepsilon\) of \(L\) on the left. The right-hand limit supplies a distance \(\delta_+>0\) with the same property on the right. Choose \(\delta=\min(\delta_-,\delta_+)\). Then every \(x\) with \(0<|x-a|<\delta\) lies on one side or the other and satisfies \(|f(x)-L|<\varepsilon\). Thus the two-sided limit is \(L\). Conversely, if the two-sided limit is \(L\), its defining condition applies to inputs on either side separately, so both one-sided limits are \(L\).
Therefore, the graph test is not a different definition of continuity. It is the continuity definition separated into facts that a graph can help you inspect: the point’s value, the nearby behavior, and whether those values agree.
Worked Example: A Filled Point Above a Hole
Worked Example: A Filled Point Above a Hole
Suppose a graph follows the line \(y=x+1\) on both sides of \(x=2\), with an open circle at \((2,3)\) and a filled dot at \((2,5)\). Decide whether the function is continuous at \(x=2\).
Step 1: Read the function value. The filled dot is at height \(5\), so \(f(2)=5\).
Step 2: Read the approach from the left. Along the line, as \(x\) approaches \(2\) from below, \(y=x+1\) approaches \(2+1=3\). Thus \(\lim_{x\to 2^-}f(x)=3\).
Step 3: Read the approach from the right. The same line continues on the right, so as \(x\) approaches \(2\) from above, \(y=x+1\) also approaches \(2+1=3\). Thus \(\lim_{x\to 2^+}f(x)=3\).
Step 4: Compare the checks. The one-sided limits agree, so the two-sided limit exists and equals \(3\). But \(f(2)=5\), and \(3\ne 5\). The function is not continuous at \(2\).
Check: The open circle marks the height approached by the line, \(3\), while the filled dot marks the actual function value, \(5\). They are different, which confirms the failed agreement condition.
This pattern is commonly called a hole or removable discontinuity: the nearby graph approaches one finite value, but the point is missing or assigned a different value. The graph test identifies the failure without requiring you to change the function.
Worked Example: Different Heights on the Two Sides
Worked Example: Different Heights on the Two Sides
Suppose the graph is a horizontal segment at height \(1\) for \(x<0\), a horizontal segment at height \(3\) for \(x>0\), and a filled dot at \((0,1)\). Decide whether the function is continuous at \(x=0\).
Step 1: Read the function value. The filled dot gives \(f(0)=1\).
Step 2: Read the left-hand limit. The graph to the left stays at height \(1\), so \(\lim_{x\to 0^-}f(x)=1\).
Step 3: Read the right-hand limit. The graph to the right stays at height \(3\), so \(\lim_{x\to 0^+}f(x)=3\).
Step 4: Test whether there is one two-sided limit. Since \(1\ne 3\), the one-sided limits do not agree. There is no two-sided limit at \(0\), so continuity fails. Although \(f(0)\) matches the left-hand limit, it cannot match a two-sided limit that does not exist.
Check: Approaching from the left gives height \(1\), while approaching from the right gives height \(3\). Following either side of the graph confirms the unequal one-sided limits.
A sudden change between two finite heights is the visual pattern of a jump. The key test is not whether the graph has a filled point at the marked input; it is whether both sides approach the same height. A defined value cannot make unequal one-sided limits agree.
Worked Example: The Graph Grows Without Bound
Worked Example: The Graph Grows Without Bound
Consider the graph of \(f(x)=\frac{1}{(x-1)^2}\), marked at \(x=1\). Decide whether the function is continuous there by inspecting its value and nearby behavior.
Step 1: Check whether the function value exists. Substituting \(x=1\) gives \(\frac{1}{(1-1)^2}=\frac{1}{0}\), which is undefined. There is no graph point at \(x=1\), so the first condition fails.
Step 2: Inspect values on either side. For \(x=0\), \(f(0)=\frac{1}{(-1)^2}=1\). For \(x=0.5\), \(f(0.5)=\frac{1}{(-0.5)^2}=4\). For \(x=1.5\), \(f(1.5)=\frac{1}{(0.5)^2}=4\), and for \(x=2\), \(f(2)=\frac{1}{1^2}=1\). The values rise as inputs on either side get closer to \(1\).
Step 3: Read the behavior as the input approaches \(1\). The squared denominator is positive for \(x\ne 1\) and gets closer to \(0\) as \(x\) approaches \(1\). Its reciprocal grows without bound on both sides. The graph does not approach one finite height, so the second condition fails too.
Step 4: Conclude and check. The function is not continuous at \(1\): it is undefined there and its nearby values grow without bound. The sample values support the graph’s upward behavior; they do not assign a finite value at \(1\) or create a finite limit.
When a graph rises or falls without bound near a marked input, it has unbounded behavior there, often shown with a vertical asymptote. This is different from a hole: a hole has a single finite height approached by the nearby graph, while unbounded behavior has no finite limiting height.
Compare the Main Graph Patterns
| Graph pattern near \(x=a\) | What to check | Continuity result |
|---|---|---|
| One curve approaches the same finite height from both sides, and the filled point is at that height. | The value exists, the limit exists, and they agree. | Continuous at \(a\). |
| The curve approaches one finite height, but the point is missing or the filled point is elsewhere. | The limit may exist, but the value is missing or does not agree. | Not continuous at \(a\). |
| The left and right sides approach different finite heights. | The one-sided limits are unequal. | Not continuous at \(a\). |
| The graph grows or falls without bound near \(a\). | There is no finite two-sided limit. | Not continuous at \(a\). |
Worked Example: A Graph That Passes All Three Checks
Worked Example: A Graph That Passes All Three Checks
Suppose the graph is \(y=|x|\), with a filled point at \((0,0)\). Decide whether it is continuous at \(x=0\).
Step 1: Read the function value. The filled point gives \(f(0)=|0|=0\).
Step 2: Follow the graph from the left. For \(x<0\), \(|x|=-x\). As \(x\) approaches \(0\) from the left, \(-x\) approaches \(0\), so \(\lim_{x\to 0^-}|x|=0\).
Step 3: Follow the graph from the right. For \(x>0\), \(|x|=x\). As \(x\) approaches \(0\) from the right, \(x\) approaches \(0\), so \(\lim_{x\to 0^+}|x|=0\).
Step 4: Compare limit and value. The one-sided limits agree, so \(\lim_{x\to 0}|x|=0\). Since \(f(0)=0\), the limit equals the function value. All three conditions hold, so the function is continuous at \(0\).
Check: Both branches meet the filled point at height \(0\); there is no missing point, jump, or unbounded behavior at the marked input.
Common Mistake: Looking Only at the Filled Dot
A filled dot tells you the function value, but it says nothing by itself about what the graph approaches nearby. For example, in the jump example, the filled dot at \((0,1)\) exists, yet the right-hand side approaches \(3\), not \(1\). Checking only \(f(0)=1\) would incorrectly suggest continuity.
The reverse mistake is to see an open circle and conclude immediately that the two-sided limit does not exist. An open circle can mark a single finite height approached from both sides, as in the first worked example. The limit can exist even though the function is not continuous there.
When Graph Reading Is Not Enough
A drawn graph is a visual representation, often at a limited scale. It can make a jump, a hole, or unbounded behavior easy to spot, but a sketch alone may not establish the exact limiting value. Two branches can look as if they meet when they do not, or a rapid oscillation can be hidden by the scale. When a formula or additional information is available, use it to verify what the picture suggests.
For a marked interior point, a reliable reading process is:
Read the height of the filled point at the marked input, if one is present.
Determine whether the graph approaches a finite height from the left and from the right.
If both sides approach one height, compare it with the function value. Continuity requires equality.
This sequence keeps the three continuity conditions separate. It also makes clear why a filled point, an open circle, a jump, or a vertical asymptote is evidence to interpret rather than a substitute for the full test.
Check Your Understanding
For each question, use the function value and the behavior from both sides of the marked input.
- A graph approaches height \(4\) from both sides of \(x=3\), but has no filled point at \(x=3\). Which continuity condition fails?
- A graph has \(f(1)=2\), while its left- and right-hand limits at \(1\) are both \(2\). Is it continuous at \(1\)? Explain using the three checks.
- A graph approaches height \(0\) from the left and height \(5\) from the right at \(x=0\). Does a two-sided limit exist?
- Near \(x=2\), a graph rises without bound on both sides. Does it approach a finite value there?
- Why is an open circle alone not enough to decide whether a two-sided limit exists?
