diff --git a/backend/app/export/exporters/html.py b/backend/app/export/exporters/html.py
index 3e3c74a..941474e 100644
--- a/backend/app/export/exporters/html.py
+++ b/backend/app/export/exporters/html.py
@@ -209,7 +209,11 @@ class HtmlExporter:
             warnings.append(self._format_plot_diagnostic(diag))
         if parsed.plot is None:
             return f'<pre class="function-plot">{html.escape(node.text)}</pre>'
-        rendered = render_svg(parsed.plot)
+        try:
+            rendered = render_svg(parsed.plot)
+        except Exception as exc:  # 渲染异常回退占位，绝不阻断整篇导出
+            warnings.append(f"函数图像：渲染失败，已回退占位（{exc}）")
+            return f'<pre class="function-plot">{html.escape(node.text)}</pre>'
         warnings.extend(rendered.warnings)
         return f'<figure class="function-plot">{rendered.content}</figure>'
 
diff --git a/backend/app/plot/parser.py b/backend/app/plot/parser.py
index fc74f9e..974b4ca 100644
--- a/backend/app/plot/parser.py
+++ b/backend/app/plot/parser.py
@@ -162,6 +162,9 @@ def _check_node(node: ast.AST) -> None:
             _unsafe(f"不支持的函数调用 {ast.dump(node.func)!r}")
         if node.keywords:
             _unsafe("函数调用不支持关键字参数")
+        # 白名单内所有函数均恰取 1 个参数，提前校验避免求值期 TypeError
+        if len(node.args) != 1:
+            _unsafe(f"{node.func.id} 需要 1 个参数，实际 {len(node.args)} 个")
         for arg in node.args:
             _check_node(arg)
         return
@@ -206,6 +209,9 @@ def _eval_node(node: ast.AST, x: float) -> float:
             return left * right
         if isinstance(node.op, ast.Div):
             return left / right
+        # 负数底 + 非整数指数会得到复数，数学绘图不支持，抛 ValueError 让采样点作为断点处理
+        if left < 0 and not right.is_integer():
+            raise ValueError("negative base with fractional exponent")
         return left**right
     if isinstance(node, ast.UnaryOp):
         value = _eval_node(node.operand, x)
diff --git a/backend/app/plot/render.py b/backend/app/plot/render.py
index 4c3dd78..0b91015 100644
--- a/backend/app/plot/render.py
+++ b/backend/app/plot/render.py
@@ -36,6 +36,8 @@ def _fmt_num(v: float) -> str:
 
 def _nice_step(span: float, target_ticks: int = 6) -> float:
     raw = abs(span) / target_ticks
+    if not math.isfinite(raw) or raw <= 0:
+        return 1.0  # 兜底步长，避免 span 为 0/inf 时产生非法刻度
     mag = 10 ** math.floor(math.log10(raw))
     for m in (1, 2, 5, 10):
         if raw <= m * mag:
@@ -44,34 +46,40 @@ def _nice_step(span: float, target_ticks: int = 6) -> float:
 
 
 def _ticks(lo: float, hi: float, step: float) -> list[float]:
+    # 防御：非法步长直接返回空，避免除零
+    if not math.isfinite(step) or step <= 0:
+        return []
     first = math.ceil(lo / step) * step
     values: list[float] = []
     v = first
-    while v <= hi + step * 1e-9:
+    # 有上限的整数索引推进 + 步长推进校验，防止浮点精度导致 v+step==v 的死循环
+    for _ in range(1000):
+        if v > hi + step * 1e-9:
+            break
         values.append(v)
-        v += step
+        nxt = v + step
+        if nxt <= v:
+            break  # 步长小于当前数值的浮点精度，已无法推进
+        v = nxt
     return values
 
 
 def _compute_range(
-    plot: FunctionPlot,
     fns: list[tuple[object, object]],
     xmin: float,
     xmax: float,
 ) -> tuple[float, float]:
-    """采样确定 y 范围；指定 range 则优先，否则取有限样本的 min/max 加 5% 余量。"""
-    if plot.range is not None:
-        return float(plot.range[0]), float(plot.range[1])
-
+    """采样确定 y 范围；取有限样本的 min/max 加 5% 余量。"""
     ys: list[float] = []
     for _expr, tree in fns:
         for i in range(_SAMPLES + 1):
             x = xmin + (xmax - xmin) * i / _SAMPLES
             try:
                 y = evaluate(tree, x)  # type: ignore[arg-type]
-            except (ValueError, ZeroDivisionError, OverflowError):
+            except (ValueError, ZeroDivisionError, OverflowError, TypeError):
                 continue
-            if math.isfinite(y):
+            # 复数等非实数结果直接跳过，不参与范围统计
+            if isinstance(y, (int, float)) and math.isfinite(y):
                 ys.append(y)
 
     if not ys:
@@ -99,9 +107,9 @@ def _polyline(
         x = xmin + (xmax - xmin) * i / _SAMPLES
         try:
             y = evaluate(tree, x)  # type: ignore[arg-type]
-        except (ValueError, ZeroDivisionError, OverflowError):
+        except (ValueError, ZeroDivisionError, OverflowError, TypeError):
             y = math.nan
-        if not math.isfinite(y):
+        if not isinstance(y, (int, float)) or not math.isfinite(y):
             if points:
                 segments.append(f'<polyline points="{" ".join(points)}" fill="none" stroke="{color}"/>')
                 points = []
@@ -178,7 +186,7 @@ def render_svg(plot: FunctionPlot) -> StaticRenderResult:
     """把已解析的 FunctionPlot 渲染为内嵌 SVG。"""
     warnings: list[str] = []
     xmin, xmax = plot.domain
-    if xmin >= xmax:
+    if not (math.isfinite(xmin) and math.isfinite(xmax)) or xmin >= xmax:
         warnings.append("domain 无效，回退到 [-10, 10]")
         xmin, xmax = -10.0, 10.0
 
@@ -192,10 +200,16 @@ def render_svg(plot: FunctionPlot) -> StaticRenderResult:
             continue
         fns.append((expr, tree))
 
-    ymin, ymax = _compute_range(plot, fns, xmin, xmax)
-    if plot.range is not None and plot.range[0] >= plot.range[1]:
-        warnings.append("range 无效，改用自动范围")
-        ymin, ymax = _compute_range(plot, fns, xmin, xmax)
+    # 纵轴范围：显式 range 有效则用之；无效（退化/非有限）丢弃并自动采样重算
+    if plot.range is not None:
+        lo, hi = float(plot.range[0]), float(plot.range[1])
+        if math.isfinite(lo) and math.isfinite(hi) and lo < hi:
+            ymin, ymax = lo, hi
+        else:
+            warnings.append("range 无效，改用自动范围")
+            ymin, ymax = _compute_range(fns, xmin, xmax)
+    else:
+        ymin, ymax = _compute_range(fns, xmin, xmax)
 
     def sx(x: float) -> float:
         return _MARGIN + (x - xmin) / (xmax - xmin) * (_WIDTH - 2 * _MARGIN)
diff --git a/backend/tests/test_plot.py b/backend/tests/test_plot.py
index ce9d727..f7ceb86 100644
--- a/backend/tests/test_plot.py
+++ b/backend/tests/test_plot.py
@@ -137,3 +137,58 @@ def test_html_exporter_function_plot_fallback_on_error() -> None:
     assert '<pre class="function-plot">' in html
     assert "<svg" not in html
     assert any("函数图像" in w for w in result.warnings)
+
+
+# --------------------------------------------------------------------------- #
+# 审阅回归：浮点刻度 / 求值异常 / 无效范围
+# --------------------------------------------------------------------------- #
+def test_render_svg_huge_domain_ticks_bounded() -> None:
+    # P1：巨大 domain 下步长受浮点精度限制无法推进，刻度应有限而非死循环
+    plot = parse_source("domain: 10000000000000000, 10000000000000002\nrange: -1, 1\ny = 0").plot
+    rendered = render_svg(plot)
+    assert "<svg" in rendered.content
+
+
+def test_parse_expression_rejects_wrong_arg_count() -> None:
+    # P2：sin() / sin(1, 2) 应在解析期拒绝，而非求值期 TypeError
+    with pytest.raises(PlotParseError):
+        parse_expression("sin()")
+    with pytest.raises(PlotParseError):
+        parse_expression("sin(1, 2)")
+
+
+def test_render_svg_nonreal_samples_are_break_points() -> None:
+    # P2：x^0.5 在负数域产生复数，应作为断点处理，正半轴仍可绘制
+    plot = parse_source("domain: -4, 4\ny = x^0.5").plot
+    rendered = render_svg(plot)
+    assert "<polyline" in rendered.content
+
+
+def test_render_svg_invalid_range_falls_back() -> None:
+    # P2：退化 range（1, 1）应丢弃并自动采样，而非 ZeroDivisionError
+    plot = parse_source("range: 1, 1\ny = x").plot
+    rendered = render_svg(plot)
+    assert "<polyline" in rendered.content
+    assert any("range" in w for w in rendered.warnings)
+
+
+def test_render_svg_nonfinite_range_falls_back() -> None:
+    # P2：非有限 range 端点应丢弃并自动采样
+    plot = parse_source("range: nan, 1\ny = x").plot
+    rendered = render_svg(plot)
+    assert "<polyline" in rendered.content
+
+
+def test_html_exporter_function_plot_render_error_falls_back(monkeypatch) -> None:
+    # P2：渲染异常不阻断整篇导出，回退占位并记 warning
+    import app.export.exporters.html as html_mod
+
+    def boom(plot):
+        raise RuntimeError("boom")
+
+    monkeypatch.setattr(html_mod, "render_svg", boom)
+    md = "```function-plot\ny = x\n```"
+    result = asyncio.run(HtmlExporter().export(parse_document(md), ExportOptions()))
+    html = result.content.decode("utf-8")
+    assert '<pre class="function-plot">' in html
+    assert any("渲染失败" in w for w in result.warnings)